Podcast
Questions and Answers
In a triangle (\triangle ABC), a line segment (DE) is drawn parallel to side (BC) and intersects sides (AB) and (AC) at points (D) and (E) respectively. If (AD = 4), (DB = 6), and (AE = 5), what is the length of (EC)?
In a triangle (\triangle ABC), a line segment (DE) is drawn parallel to side (BC) and intersects sides (AB) and (AC) at points (D) and (E) respectively. If (AD = 4), (DB = 6), and (AE = 5), what is the length of (EC)?
What is the equation of a circle with center at the origin and radius r?
What is the equation of a circle with center at the origin and radius r?
What is the symmetry of a circle with center at the origin?
What is the symmetry of a circle with center at the origin?
What is the equation of a circle with center at (a, b) and radius r?
What is the equation of a circle with center at (a, b) and radius r?
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What is the purpose of completing the square in the context of circle equations?
What is the purpose of completing the square in the context of circle equations?
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What is the general form of a circle's equation?
What is the general form of a circle's equation?
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What is the first step in completing the square to find the center and radius of a circle?
What is the first step in completing the square to find the center and radius of a circle?
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What is the equation derived from the Pythagorean theorem?
What is the equation derived from the Pythagorean theorem?
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What is the relationship between the gradient of the radius and the gradient of the tangent?
What is the relationship between the gradient of the radius and the gradient of the tangent?
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What is the purpose of using the distance formula in the context of circle equations?
What is the purpose of using the distance formula in the context of circle equations?
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What is the standard form of the equation of a circle?
What is the standard form of the equation of a circle?
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What is the key concept in determining the equation of a tangent to a circle?
What is the key concept in determining the equation of a tangent to a circle?
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What is the property of proportion that states wx = yz?
What is the property of proportion that states wx = yz?
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What is the theorem that states that a line parallel to one side of a triangle divides the other two sides proportionally?
What is the theorem that states that a line parallel to one side of a triangle divides the other two sides proportionally?
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What is the ratio of the sides of a triangle when a line is drawn parallel to one side?
What is the ratio of the sides of a triangle when a line is drawn parallel to one side?
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What is the first step in determining the equation of a tangent to a circle?
What is the first step in determining the equation of a tangent to a circle?
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What is the gradient-point form of the equation of a straight line?
What is the gradient-point form of the equation of a straight line?
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What is the property of proportion that states x/w = z/y?
What is the property of proportion that states x/w = z/y?
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What is the key concept in solving proportional problems?
What is the key concept in solving proportional problems?
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Which of the following expressions is equivalent to (\cos(\alpha + \beta)) using only the cosine difference formula and the even-odd identities?
Which of the following expressions is equivalent to (\cos(\alpha + \beta)) using only the cosine difference formula and the even-odd identities?
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In deriving the cosine difference formula, we equate the square of the distance between points K and L on the unit circle, derived using the distance formula and the cosine rule. What is the expression for the square of the distance between points K and L, using the cosine rule?
In deriving the cosine difference formula, we equate the square of the distance between points K and L on the unit circle, derived using the distance formula and the cosine rule. What is the expression for the square of the distance between points K and L, using the cosine rule?
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Using the compound angle identities, find the exact value of (\sin 15°) in terms of radicals.
Using the compound angle identities, find the exact value of (\sin 15°) in terms of radicals.
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Which of the following expressions is equivalent to (\sin(\alpha + \beta)) in terms of (\cos(\alpha - \beta)) and the even-odd identities?
Which of the following expressions is equivalent to (\sin(\alpha + \beta)) in terms of (\cos(\alpha - \beta)) and the even-odd identities?
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Which of the following is NOT a valid compound angle identity?
Which of the following is NOT a valid compound angle identity?
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If (\cos(\alpha - \beta) = \frac{1}{2}) and (\sin(\alpha - \beta) = \frac{\sqrt{3}}{2}), what is the value of (\tan(\alpha - \beta))?
If (\cos(\alpha - \beta) = \frac{1}{2}) and (\sin(\alpha - \beta) = \frac{\sqrt{3}}{2}), what is the value of (\tan(\alpha - \beta))?
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The cosine difference formula can be used to prove the cosine sum formula by rewriting the angle (\alpha + \beta)) as a difference. What is this equivalent difference?
The cosine difference formula can be used to prove the cosine sum formula by rewriting the angle (\alpha + \beta)) as a difference. What is this equivalent difference?
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Given that (\cos \alpha = \frac{3}{5}) and (\sin \beta = \frac{12}{13}), where (0° < \alpha < 90°) and (90° < \beta < 180°), what is the value of (\cos(\alpha + \beta))?
Given that (\cos \alpha = \frac{3}{5}) and (\sin \beta = \frac{12}{13}), where (0° < \alpha < 90°) and (90° < \beta < 180°), what is the value of (\cos(\alpha + \beta))?
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What relationship holds true if two triangles have their corresponding sides in proportion?
What relationship holds true if two triangles have their corresponding sides in proportion?
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If triangle ABC has angles A, B, and C, and triangle DEF has angles D, E, and F, which of the following statements is true if the triangles are similar?
If triangle ABC has angles A, B, and C, and triangle DEF has angles D, E, and F, which of the following statements is true if the triangles are similar?
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What can be concluded about the areas of two triangles with equal bases situated between the same parallel lines?
What can be concluded about the areas of two triangles with equal bases situated between the same parallel lines?
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In the proof of the similarity of triangles using the proportionality theorem, what role does the line GH serve?
In the proof of the similarity of triangles using the proportionality theorem, what role does the line GH serve?
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What is indicated by the equality of the proportions $rac{AB}{DE} = rac{AC}{DF} = rac{BC}{EF}$?
What is indicated by the equality of the proportions $rac{AB}{DE} = rac{AC}{DF} = rac{BC}{EF}$?
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If the statement of the Converse of the Pythagorean Theorem is true, what can be inferred about the triangle?
If the statement of the Converse of the Pythagorean Theorem is true, what can be inferred about the triangle?
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Which of the following correctly summarizes the key concept of equiangular triangles?
Which of the following correctly summarizes the key concept of equiangular triangles?
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Which formula must be applied to find the area of a triangle given the base and the height?
Which formula must be applied to find the area of a triangle given the base and the height?
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In proving that triangles are similar, what geometric criterion is used when establishing that corresponding angles are equal?
In proving that triangles are similar, what geometric criterion is used when establishing that corresponding angles are equal?
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How does the Pythagorean theorem relate to right-angled triangles?
How does the Pythagorean theorem relate to right-angled triangles?
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What is the simplified result of ( \sin(\alpha - \beta) ) using co-function identities?
What is the simplified result of ( \sin(\alpha - \beta) ) using co-function identities?
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Which formula represents the cosine of a sum?
Which formula represents the cosine of a sum?
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Which identity corresponds to the sine of a double angle?
Which identity corresponds to the sine of a double angle?
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What is the equivalent form of the cosine of a double angle if ( \alpha = 30^ ext{circ} )?
What is the equivalent form of the cosine of a double angle if ( \alpha = 30^ ext{circ} )?
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What is NOT a step in finding general solutions for trigonometric equations?
What is NOT a step in finding general solutions for trigonometric equations?
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In the derivation of the cosine of a double angle, what is the initial formula used?
In the derivation of the cosine of a double angle, what is the initial formula used?
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What is the correct expression for ( \cos(2\alpha) ) using only sine?
What is the correct expression for ( \cos(2\alpha) ) using only sine?
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Which compound angle formula for sine is correct for ( \alpha + \beta )?
Which compound angle formula for sine is correct for ( \alpha + \beta )?
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Which of the following correctly describes how to use a CAST diagram?
Which of the following correctly describes how to use a CAST diagram?
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Triangle ABC has a base BC and height h. A line is drawn parallel to BC, intersecting AB and AC at points D and E respectively. If the area of triangle ADE is half the area of triangle ABC, what is the ratio of AD to AB?
Triangle ABC has a base BC and height h. A line is drawn parallel to BC, intersecting AB and AC at points D and E respectively. If the area of triangle ADE is half the area of triangle ABC, what is the ratio of AD to AB?
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In triangle ABC, D and E are the midpoints of sides AB and AC respectively. If line DE is extended to meet BC at point F, which of the following statements is true?
In triangle ABC, D and E are the midpoints of sides AB and AC respectively. If line DE is extended to meet BC at point F, which of the following statements is true?
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Two triangles ABC and DEF are similar. If AB = 6, BC = 8, DE = 9, and EF = 12, what is the ratio of the area of triangle ABC to the area of triangle DEF?
Two triangles ABC and DEF are similar. If AB = 6, BC = 8, DE = 9, and EF = 12, what is the ratio of the area of triangle ABC to the area of triangle DEF?
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In triangle ABC, point D lies on side BC. If AD bisects angle BAC and BD = 3, DC = 5, what is the ratio of AB to AC?
In triangle ABC, point D lies on side BC. If AD bisects angle BAC and BD = 3, DC = 5, what is the ratio of AB to AC?
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In a triangle ABC, line segment DE is parallel to side BC and intersects sides AB and AC at points D and E respectively. If AD = 2, DB = 3, and AE = 4, what is the length of EC?
In a triangle ABC, line segment DE is parallel to side BC and intersects sides AB and AC at points D and E respectively. If AD = 2, DB = 3, and AE = 4, what is the length of EC?
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Two triangles are similar, and the ratio of their corresponding sides is 2:3. If the area of the smaller triangle is 16 square units, what is the area of the larger triangle?
Two triangles are similar, and the ratio of their corresponding sides is 2:3. If the area of the smaller triangle is 16 square units, what is the area of the larger triangle?
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In triangle ABC, D is a point on side BC such that BD = DC. If AD is perpendicular to BC, which of the following statements is true?
In triangle ABC, D is a point on side BC such that BD = DC. If AD is perpendicular to BC, which of the following statements is true?
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In triangle ABC, points D and E are the midpoints of sides AB and AC respectively. If DE = 5, what is the length of BC?
In triangle ABC, points D and E are the midpoints of sides AB and AC respectively. If DE = 5, what is the length of BC?
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Two triangles are similar, and the ratio of their perimeters is 3:4. If the area of the smaller triangle is 18 square units, what is the area of the larger triangle?
Two triangles are similar, and the ratio of their perimeters is 3:4. If the area of the smaller triangle is 18 square units, what is the area of the larger triangle?
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In triangle ABC, D is a point on side BC such that BD:DC = 2:3. If AD is an angle bisector of angle BAC, what is the ratio of AB to AC?
In triangle ABC, D is a point on side BC such that BD:DC = 2:3. If AD is an angle bisector of angle BAC, what is the ratio of AB to AC?
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A pole of unknown height stands vertically on a horizontal plane. An observer at a point (A) on the plane, (d) meters from the base of the pole, observes the top of the pole at an angle of elevation (\alpha). From a point (B), which is (b) meters closer to the pole than (A), the angle of elevation is (\beta). Express the height of the pole, (h), in terms of (d), (b), (\alpha), and (\beta).
A pole of unknown height stands vertically on a horizontal plane. An observer at a point (A) on the plane, (d) meters from the base of the pole, observes the top of the pole at an angle of elevation (\alpha). From a point (B), which is (b) meters closer to the pole than (A), the angle of elevation is (\beta). Express the height of the pole, (h), in terms of (d), (b), (\alpha), and (\beta).
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A building of unknown height stands vertically on a horizontal plane. From a point (A) on the plane, the angle of elevation to the top of the building is (\alpha). From a point (B) on the plane, (b) meters closer to the building than (A), the angle of elevation is (\beta). The angle between (A), (B), and the base of the building is (\theta). Express the height of the building, (h), in terms of (b), (\alpha), (\beta), and (\theta).
A building of unknown height stands vertically on a horizontal plane. From a point (A) on the plane, the angle of elevation to the top of the building is (\alpha). From a point (B) on the plane, (b) meters closer to the building than (A), the angle of elevation is (\beta). The angle between (A), (B), and the base of the building is (\theta). Express the height of the building, (h), in terms of (b), (\alpha), (\beta), and (\theta).
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A triangle (ABC) has sides (a), (b), and (c) opposite to angles (A), (B), and (C) respectively. If (a = 5), (b = 7), and (C = 60^\circ), calculate the area of the triangle.
A triangle (ABC) has sides (a), (b), and (c) opposite to angles (A), (B), and (C) respectively. If (a = 5), (b = 7), and (C = 60^\circ), calculate the area of the triangle.
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Two ships, (A) and (B), leave a port at the same time. Ship (A) sails on a bearing of (040^\circ) at (15) knots. Ship (B) sails on a bearing of (100^\circ) at (20) knots. After (3) hours, find the distance between the two ships.
Two ships, (A) and (B), leave a port at the same time. Ship (A) sails on a bearing of (040^\circ) at (15) knots. Ship (B) sails on a bearing of (100^\circ) at (20) knots. After (3) hours, find the distance between the two ships.
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A triangular field (ABC) has sides (a = 100) meters, (b = 150) meters, and (c = 200) meters. Calculate the measure of angle (B).
A triangular field (ABC) has sides (a = 100) meters, (b = 150) meters, and (c = 200) meters. Calculate the measure of angle (B).
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A triangle (ABC) has sides (a = 5), (b = 7), and (c = 8). Calculate the measure of angle (A).
A triangle (ABC) has sides (a = 5), (b = 7), and (c = 8). Calculate the measure of angle (A).
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A ship sails from point (A) to point (B) on a bearing of (030^\circ) for (10) nautical miles. It then sails from point (B) to point (C) on a bearing of (120^\circ) for (15) nautical miles. Calculate the distance between points (A) and (C).
A ship sails from point (A) to point (B) on a bearing of (030^\circ) for (10) nautical miles. It then sails from point (B) to point (C) on a bearing of (120^\circ) for (15) nautical miles. Calculate the distance between points (A) and (C).
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A triangle (ABC) has sides (a = 6), (b = 8), and (c = 10). Calculate the area of the triangle.
A triangle (ABC) has sides (a = 6), (b = 8), and (c = 10). Calculate the area of the triangle.
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In triangle ABC, D is a point on side BC such that AD bisects angle BAC. If AB = 8, AC = 12, and BD = 4, what is the length of DC?
In triangle ABC, D is a point on side BC such that AD bisects angle BAC. If AB = 8, AC = 12, and BD = 4, what is the length of DC?
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Given a circle with the equation ( (x - 2)^2 + (y + 3)^2 = 16 ), what is the gradient of the tangent line at the point ( (6, 1) )?
Given a circle with the equation ( (x - 2)^2 + (y + 3)^2 = 16 ), what is the gradient of the tangent line at the point ( (6, 1) )?
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A line is drawn parallel to one side of a triangle, dividing the other two sides proportionally. If the line divides one side into segments of lengths 5 and 8, and the corresponding segment on the other side is 12, what is the length of the other segment on that side?
A line is drawn parallel to one side of a triangle, dividing the other two sides proportionally. If the line divides one side into segments of lengths 5 and 8, and the corresponding segment on the other side is 12, what is the length of the other segment on that side?
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The equation of a circle is ( x^2 + y^2 - 6x + 8y = 0 ). What is the radius of the circle?
The equation of a circle is ( x^2 + y^2 - 6x + 8y = 0 ). What is the radius of the circle?
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A circle has a center at ( (3, -2) ) and a radius of ( 5 ). What is the equation of the tangent line at the point ( (7, -2) )?
A circle has a center at ( (3, -2) ) and a radius of ( 5 ). What is the equation of the tangent line at the point ( (7, -2) )?
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The equation of a circle is ( (x + 1)^2 + (y - 4)^2 = 9 ). What are the coordinates of the center of the circle?
The equation of a circle is ( (x + 1)^2 + (y - 4)^2 = 9 ). What are the coordinates of the center of the circle?
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In a triangle ( ABC ), a line segment ( DE ) is drawn parallel to side ( BC ) and intersects sides ( AB ) and ( AC ) at points ( D ) and ( E ), respectively. If ( AD = 3 ), ( DB = 5 ), and ( AE = 4 ), what is the length of ( EC )?
In a triangle ( ABC ), a line segment ( DE ) is drawn parallel to side ( BC ) and intersects sides ( AB ) and ( AC ) at points ( D ) and ( E ), respectively. If ( AD = 3 ), ( DB = 5 ), and ( AE = 4 ), what is the length of ( EC )?
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Two ratios, ( a/b ) and ( c/d ), are proportional. Which of the following expressions is NOT necessarily true?
Two ratios, ( a/b ) and ( c/d ), are proportional. Which of the following expressions is NOT necessarily true?
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A circle has a center at ( (1, -2) ) and passes through the point ( (4, 1) ). What is the equation of the circle?
A circle has a center at ( (1, -2) ) and passes through the point ( (4, 1) ). What is the equation of the circle?
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A tangent line to a circle with center ( (2, 3) ) passes through the point ( (5, 1) ). What is the slope of the tangent line?
A tangent line to a circle with center ( (2, 3) ) passes through the point ( (5, 1) ). What is the slope of the tangent line?
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Two triangles, ( ABC ) and ( DEF ), are similar. If ( AB = 6 ), ( BC = 8 ), and ( DE = 9 ), what is the length of ( EF )?
Two triangles, ( ABC ) and ( DEF ), are similar. If ( AB = 6 ), ( BC = 8 ), and ( DE = 9 ), what is the length of ( EF )?
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In triangle ABC, a line segment DE is drawn parallel to side BC and intersects sides AB and AC at points D and E respectively. If AD = 4, DB = 6, and AE = 5, what is the length of EC?
In triangle ABC, a line segment DE is drawn parallel to side BC and intersects sides AB and AC at points D and E respectively. If AD = 4, DB = 6, and AE = 5, what is the length of EC?
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Two triangles ABC and DEF are similar. If AB = 6, BC = 8, DE = 9, and EF = 12, what is the ratio of the area of triangle ABC to the area of triangle DEF?
Two triangles ABC and DEF are similar. If AB = 6, BC = 8, DE = 9, and EF = 12, what is the ratio of the area of triangle ABC to the area of triangle DEF?
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In triangle ABC, point D lies on side BC. If AD bisects angle BAC and BD = 3, DC = 5, what is the ratio of AB to AC?
In triangle ABC, point D lies on side BC. If AD bisects angle BAC and BD = 3, DC = 5, what is the ratio of AB to AC?
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Triangle ABC has a base BC and height h. A line is drawn parallel to BC, intersecting AB and AC at points D and E respectively. If the area of triangle ADE is half the area of triangle ABC, what is the ratio of AD to AB?
Triangle ABC has a base BC and height h. A line is drawn parallel to BC, intersecting AB and AC at points D and E respectively. If the area of triangle ADE is half the area of triangle ABC, what is the ratio of AD to AB?
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In triangle ABC, D and E are the midpoints of sides AB and AC respectively. If line DE is extended to meet BC at point F, which of the following statements is true?
In triangle ABC, D and E are the midpoints of sides AB and AC respectively. If line DE is extended to meet BC at point F, which of the following statements is true?
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In the proof of the theorem "Triangles with Sides in Proportion are Similar", which geometric concept is employed to demonstrate that (GH \parallel BC)?
In the proof of the theorem "Triangles with Sides in Proportion are Similar", which geometric concept is employed to demonstrate that (GH \parallel BC)?
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Which of the following correctly summarizes the key concept of equiangular triangles?
Which of the following correctly summarizes the key concept of equiangular triangles?
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In the proof of the Pythagorean theorem, which similarity criterion is used to establish the similarity between ( riangle ABD) and ( riangle CBA)?
In the proof of the Pythagorean theorem, which similarity criterion is used to establish the similarity between ( riangle ABD) and ( riangle CBA)?
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In proving that triangles are similar, what geometric criterion is used when establishing that corresponding angles are equal?
In proving that triangles are similar, what geometric criterion is used when establishing that corresponding angles are equal?
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What is the key concept used to derive the equation (AB^2 = BD \cdot BC) in the proof of the Pythagorean theorem?
What is the key concept used to derive the equation (AB^2 = BD \cdot BC) in the proof of the Pythagorean theorem?
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What is the simplified result of ( \sin(\alpha - \beta) ) using co-function identities?
What is the simplified result of ( \sin(\alpha - \beta) ) using co-function identities?
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In the proof of the Converse of the Pythagorean Theorem, which statement is used to infer that (\angle A) is a right angle?
In the proof of the Converse of the Pythagorean Theorem, which statement is used to infer that (\angle A) is a right angle?
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Which of the following correctly describes how to use a CAST diagram?
Which of the following correctly describes how to use a CAST diagram?
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Which of the following statements accurately describes the relationship between the areas of two triangles with equal bases situated between the same parallel lines?
Which of the following statements accurately describes the relationship between the areas of two triangles with equal bases situated between the same parallel lines?
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Which formula represents the cosine of a sum?
Which formula represents the cosine of a sum?
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What is the fundamental concept that underlies the relationship between the sides and angles of equiangular triangles?
What is the fundamental concept that underlies the relationship between the sides and angles of equiangular triangles?
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Given two triangles ( riangle ABC) and ( riangle DEF) with (rac{AB}{DE} = rac{AC}{DF} = rac{BC}{EF}), what can be concluded about the angles of the triangles?
Given two triangles ( riangle ABC) and ( riangle DEF) with (rac{AB}{DE} = rac{AC}{DF} = rac{BC}{EF}), what can be concluded about the angles of the triangles?
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In the proof of the theorem "Triangles with Sides in Proportion are Similar", what is the purpose of constructing the line segment (GH) such that (AG = DE) and (AH = DF)?
In the proof of the theorem "Triangles with Sides in Proportion are Similar", what is the purpose of constructing the line segment (GH) such that (AG = DE) and (AH = DF)?
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Which of the following statements is TRUE if the square of one side of a triangle is equal to the sum of the squares of the other two sides?
Which of the following statements is TRUE if the square of one side of a triangle is equal to the sum of the squares of the other two sides?
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In the proof of the Pythagorean theorem, what is the key role of the line segment (AD) drawn perpendicular to (BC)?
In the proof of the Pythagorean theorem, what is the key role of the line segment (AD) drawn perpendicular to (BC)?
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What is the formula for the area of a triangle when no perpendicular height is given?
What is the formula for the area of a triangle when no perpendicular height is given?
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When should the sine rule be used?
When should the sine rule be used?
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What is the formula for the height of a pole?
What is the formula for the height of a pole?
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What is the formula for the height of a building?
What is the formula for the height of a building?
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What is the first step in solving problems in three dimensions?
What is the first step in solving problems in three dimensions?
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What is the purpose of the cosine rule?
What is the purpose of the cosine rule?
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What is the formula fortan β in the problem of the height of a pole?
What is the formula fortan β in the problem of the height of a pole?
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What is the strategy for solving trigonometric problems in three dimensions?
What is the strategy for solving trigonometric problems in three dimensions?
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What is the equation of a circle with center at (2, -3) and radius 5?
What is the equation of a circle with center at (2, -3) and radius 5?
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What is the center and radius of the circle represented by the equation x^2 + y^2 - 6x + 4y - 12 = 0?
What is the center and radius of the circle represented by the equation x^2 + y^2 - 6x + 4y - 12 = 0?
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The equation of a circle is given as (x - 3)^2 + (y + 2)^2 = 16. Which of the following points lies on the circle?
The equation of a circle is given as (x - 3)^2 + (y + 2)^2 = 16. Which of the following points lies on the circle?
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Which of the following is NOT a property of a circle with center at the origin?
Which of the following is NOT a property of a circle with center at the origin?
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What is the equation of the circle with center at (-1, 4) and passing through the point (2, 1)?
What is the equation of the circle with center at (-1, 4) and passing through the point (2, 1)?
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If the equation of a circle is x^2 + y^2 + 8x - 10y + 25 = 0, what is the value of the radius?
If the equation of a circle is x^2 + y^2 + 8x - 10y + 25 = 0, what is the value of the radius?
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A circle is tangent to both the x-axis and the y-axis. If the circle passes through the point (4, 4), what is the equation of the circle?
A circle is tangent to both the x-axis and the y-axis. If the circle passes through the point (4, 4), what is the equation of the circle?
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A circle with center at (h, k) and radius r is represented by the equation (x - h)^2 + (y - k)^2 = r^2. What is the effect on the equation if the radius is doubled?
A circle with center at (h, k) and radius r is represented by the equation (x - h)^2 + (y - k)^2 = r^2. What is the effect on the equation if the radius is doubled?
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What is the compound angle formula for the sine of the sum of two angles?
What is the compound angle formula for the sine of the sum of two angles?
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Using the co-function identities, what is the correct expression for $ an(90^ ext{circ} - heta)$?
Using the co-function identities, what is the correct expression for $ an(90^ ext{circ} - heta)$?
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Which of the following represents the cosine of a difference?
Which of the following represents the cosine of a difference?
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Which alternate form of the cosine double angle formula is derived from the identity $sin^2 heta + cos^2 heta = 1$?
Which alternate form of the cosine double angle formula is derived from the identity $sin^2 heta + cos^2 heta = 1$?
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What is the correct general solution to the equation $sin(x) = 0$?
What is the correct general solution to the equation $sin(x) = 0$?
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From the double angle identity, what is the equivalent expression for $sin(2 heta)$?
From the double angle identity, what is the equivalent expression for $sin(2 heta)$?
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Which of these options correctly describes the component identities for sine based on co-function identities?
Which of these options correctly describes the component identities for sine based on co-function identities?
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In solving the equation $cos(x) = rac{1}{2}$, which of the following represents the solution in the general form?
In solving the equation $cos(x) = rac{1}{2}$, which of the following represents the solution in the general form?
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Which compound angle formula correctly represents $ an( heta + eta)$?
Which compound angle formula correctly represents $ an( heta + eta)$?
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Using the compound angle identities, find the exact value of (\cos 15^\circ) in terms of radicals.
Using the compound angle identities, find the exact value of (\cos 15^\circ) in terms of radicals.
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Which of the following expressions is equivalent to (\sin(\alpha + \beta)) in terms of (\cos(\alpha - \beta)) and the even-odd identities?
Which of the following expressions is equivalent to (\sin(\alpha + \beta)) in terms of (\cos(\alpha - \beta)) and the even-odd identities?
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If (\cos(\alpha - \beta) = \frac{1}{2}) and (\sin(\alpha - \beta) = \frac{\sqrt{3}}{2}), what is the value of (\tan(\alpha - \beta)?
If (\cos(\alpha - \beta) = \frac{1}{2}) and (\sin(\alpha - \beta) = \frac{\sqrt{3}}{2}), what is the value of (\tan(\alpha - \beta)?
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The cosine difference formula can be used to prove the cosine sum formula by rewriting the angle (\alpha + \beta)) as a difference. What is this equivalent difference?
The cosine difference formula can be used to prove the cosine sum formula by rewriting the angle (\alpha + \beta)) as a difference. What is this equivalent difference?
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Given that (\cos \alpha = \frac{3}{5}) and (\sin \beta = \frac{12}{13}), where (0^\circ < \alpha < 90^\circ) and (90^\circ < \beta < 180^\circ), what is the value of (\cos(\alpha + \beta))?
Given that (\cos \alpha = \frac{3}{5}) and (\sin \beta = \frac{12}{13}), where (0^\circ < \alpha < 90^\circ) and (90^\circ < \beta < 180^\circ), what is the value of (\cos(\alpha + \beta))?
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Which of the following correctly summarizes the key concept of equiangular triangles?
Which of the following correctly summarizes the key concept of equiangular triangles?
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In proving that triangles are similar, what geometric criterion is used when establishing that corresponding angles are equal?
In proving that triangles are similar, what geometric criterion is used when establishing that corresponding angles are equal?
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What is the simplified result of (\sin(\alpha - \beta)) using co-function identities?
What is the simplified result of (\sin(\alpha - \beta)) using co-function identities?
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What is the geometric concept that relates to the equality of ratios of corresponding sides in two triangles?
What is the geometric concept that relates to the equality of ratios of corresponding sides in two triangles?
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What is the condition for two ratios to be in proportion?
What is the condition for two ratios to be in proportion?
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What is the property of proportion that states wx = yz?
What is the property of proportion that states wx = yz?
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What is the equation of a tangent to a circle at the point of tangency (x1, y1)?
What is the equation of a tangent to a circle at the point of tangency (x1, y1)?
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What is the relationship between the gradient of the radius and the gradient of the tangent?
What is the relationship between the gradient of the radius and the gradient of the tangent?
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What is the first step in determining the equation of a tangent to a circle?
What is the first step in determining the equation of a tangent to a circle?
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What is the key concept in solving proportional problems?
What is the key concept in solving proportional problems?
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What is the theorem that states a line parallel to one side of a triangle divides the other two sides proportionally?
What is the theorem that states a line parallel to one side of a triangle divides the other two sides proportionally?
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What is the ratio of the sides of a triangle when a line is drawn parallel to one side?
What is the ratio of the sides of a triangle when a line is drawn parallel to one side?
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What is the application of proportions in geometric figures?
What is the application of proportions in geometric figures?
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What is the equivalent form of the sine of a difference formula using co-function identities?
What is the equivalent form of the sine of a difference formula using co-function identities?
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Which of the following is a correct step in finding general solutions for trigonometric equations?
Which of the following is a correct step in finding general solutions for trigonometric equations?
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What is the correct expression for cos(2α) using only sine?
What is the correct expression for cos(2α) using only sine?
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Which compound angle formula for cosine is correct for α - β?
Which compound angle formula for cosine is correct for α - β?
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What is the purpose of using the CAST diagram in solving trigonometric equations?
What is the purpose of using the CAST diagram in solving trigonometric equations?
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Which of the following is NOT a step in deriving the cosine of a double angle?
Which of the following is NOT a step in deriving the cosine of a double angle?
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What is the correct expression for sin(2α) using cosine?
What is the correct expression for sin(2α) using cosine?
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Which of the following correctly describes the sine of a double angle?
Which of the following correctly describes the sine of a double angle?
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What is the equivalent form of the cosine of a double angle if α = 30°?
What is the equivalent form of the cosine of a double angle if α = 30°?
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Which of the following is a correct step in deriving the sine of a double angle?
Which of the following is a correct step in deriving the sine of a double angle?
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What is the correct form of the equation for a circle with center at (a, b) and radius r?
What is the correct form of the equation for a circle with center at (a, b) and radius r?
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In completing the square for the equation $x^2 + y^2 + Dx + Ey + F = 0$, which term is added during the first step for the x-terms?
In completing the square for the equation $x^2 + y^2 + Dx + Ey + F = 0$, which term is added during the first step for the x-terms?
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Using the compound angle identities, what is the equivalent expression for (\cos(\alpha - \beta)) in terms of (\cos(\alpha + \beta)) and the even-odd identities?
Using the compound angle identities, what is the equivalent expression for (\cos(\alpha - \beta)) in terms of (\cos(\alpha + \beta)) and the even-odd identities?
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What represents the distance from the origin to a point P(x, y) on a circle with radius r?
What represents the distance from the origin to a point P(x, y) on a circle with radius r?
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If (\cos(\alpha - \beta) = \frac{1}{2}) and (\sin(\alpha - \beta) = \frac{\sqrt{3}}{2}), what is the value of (\tan(\alpha - \beta))?
If (\cos(\alpha - \beta) = \frac{1}{2}) and (\sin(\alpha - \beta) = \frac{\sqrt{3}}{2}), what is the value of (\tan(\alpha - \beta))?
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Which statement accurately reflects the symmetry properties of a circle centered at the origin?
Which statement accurately reflects the symmetry properties of a circle centered at the origin?
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Which of the following is NOT a valid compound angle identity?
Which of the following is NOT a valid compound angle identity?
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Which equation correctly represents the relationship established by squaring the distance formula for a radius r in terms of x and y coordinates on a circle?
Which equation correctly represents the relationship established by squaring the distance formula for a radius r in terms of x and y coordinates on a circle?
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The cosine difference formula can be used to prove the cosine sum formula by rewriting the angle (\alpha + \beta) as a difference. What is this equivalent difference?
The cosine difference formula can be used to prove the cosine sum formula by rewriting the angle (\alpha + \beta) as a difference. What is this equivalent difference?
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When deriving the standard equation of a circle from the general form $x^2 + y^2 + Dx + Ey + F = 0$, what must be true about D and E?
When deriving the standard equation of a circle from the general form $x^2 + y^2 + Dx + Ey + F = 0$, what must be true about D and E?
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Given that (\cos\alpha = \frac{3}{5}) and (\sin\beta = \frac{12}{13}), where (0° < \alpha < 90°) and (90° < \beta < 180°), what is the value of (\cos(\alpha + \beta))?
Given that (\cos\alpha = \frac{3}{5}) and (\sin\beta = \frac{12}{13}), where (0° < \alpha < 90°) and (90° < \beta < 180°), what is the value of (\cos(\alpha + \beta))?
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What relationship holds true if two triangles have their corresponding sides in proportion?
What relationship holds true if two triangles have their corresponding sides in proportion?
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After completing the square, what is the general transformation for the equation of a circle?
After completing the square, what is the general transformation for the equation of a circle?
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What is the role of the Pythagorean theorem in deriving the equation of a circle?
What is the role of the Pythagorean theorem in deriving the equation of a circle?
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Which of the following correctly summarizes the key concept of equiangular triangles?
Which of the following correctly summarizes the key concept of equiangular triangles?
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In proving that triangles are similar, what geometric criterion is used when establishing that corresponding angles are equal?
In proving that triangles are similar, what geometric criterion is used when establishing that corresponding angles are equal?
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In triangle ABC, a line segment DE is drawn parallel to side BC, intersecting sides AB and AC at points D and E respectively. If AD = 4, DB = 6, and AE = 5, what is the length of EC?
In triangle ABC, a line segment DE is drawn parallel to side BC, intersecting sides AB and AC at points D and E respectively. If AD = 4, DB = 6, and AE = 5, what is the length of EC?
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In the proof of the proportionality theorem, why is line segment GH constructed parallel to BC?
In the proof of the proportionality theorem, why is line segment GH constructed parallel to BC?
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In the proof of the Pythagorean Theorem, how does the similarity of triangles ABD and CBA contribute to the derivation of the equation (AB^2 = BD \cdot BC)?
In the proof of the Pythagorean Theorem, how does the similarity of triangles ABD and CBA contribute to the derivation of the equation (AB^2 = BD \cdot BC)?
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What is the primary implication of the statement (rac{AB}{DE} = rac{AC}{DF} = rac{BC}{EF}) in relation to triangles ABC and DEF?
What is the primary implication of the statement (rac{AB}{DE} = rac{AC}{DF} = rac{BC}{EF}) in relation to triangles ABC and DEF?
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Which of the following statements is NOT a valid condition for determining similarity of triangles?
Which of the following statements is NOT a valid condition for determining similarity of triangles?
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In the context of proportionality in triangles, what does the phrase "triangles with equal bases between the same parallel lines are equal in area" imply?
In the context of proportionality in triangles, what does the phrase "triangles with equal bases between the same parallel lines are equal in area" imply?
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What is the main difference between the Pythagorean Theorem and its converse?
What is the main difference between the Pythagorean Theorem and its converse?
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In the proof of the Pythagorean Theorem, why is it important to establish that (\angle A_1 = \angle C) and (\angle B = \angle A_2)?
In the proof of the Pythagorean Theorem, why is it important to establish that (\angle A_1 = \angle C) and (\angle B = \angle A_2)?
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Which of the following is NOT a direct consequence of the proportionality theorem?
Which of the following is NOT a direct consequence of the proportionality theorem?
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What is the key concept that distinguishes the similarity of triangles from their congruence?
What is the key concept that distinguishes the similarity of triangles from their congruence?
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In the proof of the Pythagorean Theorem, why is it necessary to construct AD perpendicular to BC?
In the proof of the Pythagorean Theorem, why is it necessary to construct AD perpendicular to BC?
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What is the relationship between the heights of two triangles on the same base and equal in area?
What is the relationship between the heights of two triangles on the same base and equal in area?
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Which of the following conditions must be satisfied for two polygons to be considered similar?
Which of the following conditions must be satisfied for two polygons to be considered similar?
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What does the Mid-point Theorem state regarding the sides of a triangle?
What does the Mid-point Theorem state regarding the sides of a triangle?
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What conclusion can be drawn from the statement 'If two triangles are equiangular, then their corresponding sides are in proportion'?
What conclusion can be drawn from the statement 'If two triangles are equiangular, then their corresponding sides are in proportion'?
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What must be true for two triangles to be proven similar by the Proportion Theorem?
What must be true for two triangles to be proven similar by the Proportion Theorem?
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Which of the following statements about the areas of triangles situated between the same parallel lines is true?
Which of the following statements about the areas of triangles situated between the same parallel lines is true?
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In the context of polygons, what does the term 'congruent' imply?
In the context of polygons, what does the term 'congruent' imply?
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What logical step is involved in proving two triangles similar using equiangular criteria?
What logical step is involved in proving two triangles similar using equiangular criteria?
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Which of the following formulas represents the relationship of areas for triangles having equal bases between two parallel lines?
Which of the following formulas represents the relationship of areas for triangles having equal bases between two parallel lines?
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In a triangle ABC, a line segment DE is drawn parallel to side BC and intersects sides AB and AC at points D and E respectively. If AD = 4, DB = 6, and AE = 5, what method should be used to find the length of EC?
In a triangle ABC, a line segment DE is drawn parallel to side BC and intersects sides AB and AC at points D and E respectively. If AD = 4, DB = 6, and AE = 5, what method should be used to find the length of EC?
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If a triangle has angles A, B, and C, and a second triangle has angles D, E, and F, what must be true for the triangles to be similar?
If a triangle has angles A, B, and C, and a second triangle has angles D, E, and F, what must be true for the triangles to be similar?
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What is the purpose of using the Sine Rule in trigonometry?
What is the purpose of using the Sine Rule in trigonometry?
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In a triangle ABC, if angle A is 30 degrees and side a = 5, what is the value of side b?
In a triangle ABC, if angle A is 30 degrees and side a = 5, what is the value of side b?
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What is the formula for the area of a triangle, given two sides and the included angle?
What is the formula for the area of a triangle, given two sides and the included angle?
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If a triangle has angles A, B, and C, and a second triangle has angles D, E, and F, what can be concluded about the triangles if the corresponding angles are equal?
If a triangle has angles A, B, and C, and a second triangle has angles D, E, and F, what can be concluded about the triangles if the corresponding angles are equal?
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What is the purpose of using the Cosine Rule in trigonometry?
What is the purpose of using the Cosine Rule in trigonometry?
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What is the formula for the height of a pole, given the distance from the observer to the base of the pole and the angle of elevation?
What is the formula for the height of a pole, given the distance from the observer to the base of the pole and the angle of elevation?
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If the equation of a circle is given as (x - a)^2 + (y - b)^2 = r^2, what is the coordinates of the center of the circle?
If the equation of a circle is given as (x - a)^2 + (y - b)^2 = r^2, what is the coordinates of the center of the circle?
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What is the relationship between the gradient of the radius and the gradient of the tangent in a circle?
What is the relationship between the gradient of the radius and the gradient of the tangent in a circle?
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If two ratios are equal, what is the relationship between the quantities involved?
If two ratios are equal, what is the relationship between the quantities involved?
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What is the key concept in determining the equation of a tangent to a circle?
What is the key concept in determining the equation of a tangent to a circle?
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What is the equation of a tangent to a circle in the gradient-point form?
What is the equation of a tangent to a circle in the gradient-point form?
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What is the property of proportion that states wx = yz?
What is the property of proportion that states wx = yz?
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What is the theorem that states that a line parallel to one side of a triangle divides the other two sides proportionally?
What is the theorem that states that a line parallel to one side of a triangle divides the other two sides proportionally?
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What is the ratio of the sides of a triangle when a line is drawn parallel to one side?
What is the ratio of the sides of a triangle when a line is drawn parallel to one side?
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What is the first step in determining the equation of a tangent to a circle?
What is the first step in determining the equation of a tangent to a circle?
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What is the purpose of using proportions in geometry?
What is the purpose of using proportions in geometry?
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If two triangles have their corresponding sides in proportion, which of the following statements is true?
If two triangles have their corresponding sides in proportion, which of the following statements is true?
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What is the purpose of Thales' theorem in geometry?
What is the purpose of Thales' theorem in geometry?
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If two triangles are equiangular, which of the following is true?
If two triangles are equiangular, which of the following is true?
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What is the name of the theorem that states that a line parallel to one side of a triangle divides the other two sides proportionally?
What is the name of the theorem that states that a line parallel to one side of a triangle divides the other two sides proportionally?
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If triangle ABC has a base BC and height h, and a line is drawn parallel to BC, intersecting AB and AC at points D and E respectively, what can be concluded about the areas of triangles ABC and ADE?
If triangle ABC has a base BC and height h, and a line is drawn parallel to BC, intersecting AB and AC at points D and E respectively, what can be concluded about the areas of triangles ABC and ADE?
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What is the name of the theorem that states that in any right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides?
What is the name of the theorem that states that in any right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides?
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If two triangles have equal bases and are situated between the same parallel lines, what can be concluded about their areas?
If two triangles have equal bases and are situated between the same parallel lines, what can be concluded about their areas?
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What is the formula for the area of a parallelogram?
What is the formula for the area of a parallelogram?
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If two triangles are similar, which of the following is true?
If two triangles are similar, which of the following is true?
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What is the purpose of the Triangle Proportionality theorem?
What is the purpose of the Triangle Proportionality theorem?
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Which of the following formulas is used to derive the sine of a double angle?
Which of the following formulas is used to derive the sine of a double angle?
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Given the equation of a circle (x^2 + y^2 - 6x + 4y - 12 = 0), what is the radius of the circle?
Given the equation of a circle (x^2 + y^2 - 6x + 4y - 12 = 0), what is the radius of the circle?
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What is the equivalent form of the cosine of a double angle if α = 45°?
What is the equivalent form of the cosine of a double angle if α = 45°?
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What is the equation of the circle with center at ((-2, 3)) and passing through the point ((1, -1))?
What is the equation of the circle with center at ((-2, 3)) and passing through the point ((1, -1))?
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Which of the following is NOT a step in finding general solutions for trigonometric equations?
Which of the following is NOT a step in finding general solutions for trigonometric equations?
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What is the initial formula used in the derivation of the cosine of a double angle?
What is the initial formula used in the derivation of the cosine of a double angle?
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If the equation of a circle is (x^2 + y^2 + 10x - 8y + 25 = 0), what is the center of the circle?
If the equation of a circle is (x^2 + y^2 + 10x - 8y + 25 = 0), what is the center of the circle?
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Which identity corresponds to the cosine of a double angle?
Which identity corresponds to the cosine of a double angle?
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Which of the following points lies on the circle with the equation (x^2 + y^2 - 8x + 6y = 0)?
Which of the following points lies on the circle with the equation (x^2 + y^2 - 8x + 6y = 0)?
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What is the correct expression for cos(2α) using only sine?
What is the correct expression for cos(2α) using only sine?
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Given that the equation of a circle is (x^2 + y^2 + 2x - 4y - 20 = 0), what is the equation of the tangent line to this circle at the point ((3, 5))?
Given that the equation of a circle is (x^2 + y^2 + 2x - 4y - 20 = 0), what is the equation of the tangent line to this circle at the point ((3, 5))?
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Which compound angle formula for sine is correct for α + β?
Which compound angle formula for sine is correct for α + β?
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The equation of a circle is (x^2 + y^2 - 10x + 8y + 16 = 0). What is the length of the diameter of this circle?
The equation of a circle is (x^2 + y^2 - 10x + 8y + 16 = 0). What is the length of the diameter of this circle?
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Which of the following correctly describes how to use a CAST diagram?
Which of the following correctly describes how to use a CAST diagram?
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The point ( (3, -1) ) lies on the circle with equation (x^2 + y^2 - 4x + 2y - 20 = 0). What is the equation of the tangent line to the circle at this point?
The point ( (3, -1) ) lies on the circle with equation (x^2 + y^2 - 4x + 2y - 20 = 0). What is the equation of the tangent line to the circle at this point?
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In a triangle, if the sine of an angle is x, what is the general solution for the angle?
In a triangle, if the sine of an angle is x, what is the general solution for the angle?
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What is the formula for the area of a triangle, given two sides and the included angle?
What is the formula for the area of a triangle, given two sides and the included angle?
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What is the simplified result of sin(α - β) using co-function identities?
What is the simplified result of sin(α - β) using co-function identities?
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What is the equation of the circle with diameter endpoints at ((2, 5)) and ((8, 1))?
What is the equation of the circle with diameter endpoints at ((2, 5)) and ((8, 1))?
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When should the cosine rule be used?
When should the cosine rule be used?
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Which formula represents the cosine of a sum?
Which formula represents the cosine of a sum?
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What is the formula for the height of a pole, given the distance from the point to the base of the pole and the angle of elevation?
What is the formula for the height of a pole, given the distance from the point to the base of the pole and the angle of elevation?
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What is the first step in solving problems in three dimensions?
What is the first step in solving problems in three dimensions?
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What is the formula for the area of a triangle, given the base and height?
What is the formula for the area of a triangle, given the base and height?
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When should the sine rule be used?
When should the sine rule be used?
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What is the formula for the height of a building, given the distance from the point to the base of the building and the angles of elevation and depression?
What is the formula for the height of a building, given the distance from the point to the base of the building and the angles of elevation and depression?
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In the proof of the Pythagorean Theorem, why is it essential to draw a perpendicular line (AD) from (A) to (BC)?
In the proof of the Pythagorean Theorem, why is it essential to draw a perpendicular line (AD) from (A) to (BC)?
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In the proof of the similarity of triangles with sides in proportion, what is the key role of the line segment (GH) constructed parallel to (BC)?
In the proof of the similarity of triangles with sides in proportion, what is the key role of the line segment (GH) constructed parallel to (BC)?
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What is the significance of the statement 'Triangles with equal heights have areas proportional to their bases' in the context of triangles and proportionality?
What is the significance of the statement 'Triangles with equal heights have areas proportional to their bases' in the context of triangles and proportionality?
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Which of the following statements is NOT a valid conclusion from the statement 'If two triangles are equiangular, the corresponding sides are in proportion, making the triangles similar.'?
Which of the following statements is NOT a valid conclusion from the statement 'If two triangles are equiangular, the corresponding sides are in proportion, making the triangles similar.'?
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In the context of the Pythagorean Theorem, what is the key relationship between the similar triangles (ABD), (CAD), and (CBA)?
In the context of the Pythagorean Theorem, what is the key relationship between the similar triangles (ABD), (CAD), and (CBA)?
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Two triangles, ( riangle ABC) and ( riangle DEF), share a common base (BC = EF). If the area of ( riangle ABC) is twice the area of ( riangle DEF), what can be concluded about the heights of these triangles?
Two triangles, ( riangle ABC) and ( riangle DEF), share a common base (BC = EF). If the area of ( riangle ABC) is twice the area of ( riangle DEF), what can be concluded about the heights of these triangles?
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What is the most accurate interpretation of the converse of the Pythagorean Theorem?
What is the most accurate interpretation of the converse of the Pythagorean Theorem?
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In a triangle ( riangle ABC), a line segment (DE) is drawn parallel to side (BC) and intersects sides (AB) and (AC) at points (D) and (E) respectively. If (AD = 4), (DB = 6), and (AE = 5), what is the length of (EC)?
In a triangle ( riangle ABC), a line segment (DE) is drawn parallel to side (BC) and intersects sides (AB) and (AC) at points (D) and (E) respectively. If (AD = 4), (DB = 6), and (AE = 5), what is the length of (EC)?
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What is the primary function of the 'SSS' similarity criterion in the proof of the Pythagorean Theorem?
What is the primary function of the 'SSS' similarity criterion in the proof of the Pythagorean Theorem?
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Which of the following concepts is NOT directly related to the concept of 'proportionality in triangles'?
Which of the following concepts is NOT directly related to the concept of 'proportionality in triangles'?
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In triangle ( riangle ABC), points (D) and (E) are the midpoints of sides (AB) and (AC), respectively. If (DE = 8), what is the length of (BC)?
In triangle ( riangle ABC), points (D) and (E) are the midpoints of sides (AB) and (AC), respectively. If (DE = 8), what is the length of (BC)?
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If two triangles have corresponding sides in proportion, what can you definitively conclude about the triangles?
If two triangles have corresponding sides in proportion, what can you definitively conclude about the triangles?
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In triangle ( riangle ABC), line segment (DE) is drawn parallel to side (BC) and intersects sides (AB) and (AC) at points (D) and (E) respectively. If (AD = 3), (DB = 5), and (AE = 4), what is the length of (EC)?
In triangle ( riangle ABC), line segment (DE) is drawn parallel to side (BC) and intersects sides (AB) and (AC) at points (D) and (E) respectively. If (AD = 3), (DB = 5), and (AE = 4), what is the length of (EC)?
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In the proof of the Pythagorean Theorem, how is the area of the larger triangle (ABC) related to the areas of the smaller triangles (ABD) and (CAD)?
In the proof of the Pythagorean Theorem, how is the area of the larger triangle (ABC) related to the areas of the smaller triangles (ABD) and (CAD)?
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If two triangles are similar, which of the following statements is ALWAYS TRUE?
If two triangles are similar, which of the following statements is ALWAYS TRUE?
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Given that ( riangle ABC \sim riangle DEF), with (AB = 5), (BC = 7), and (DF = 10), what is the length of (EF)?
Given that ( riangle ABC \sim riangle DEF), with (AB = 5), (BC = 7), and (DF = 10), what is the length of (EF)?
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In a triangle ( riangle ABC), (D) is the midpoint of side (AB), and (E) is the midpoint of side (AC). If (DE = 6) and (BC = 10), what is the relationship between (DE) and (BC)?
In a triangle ( riangle ABC), (D) is the midpoint of side (AB), and (E) is the midpoint of side (AC). If (DE = 6) and (BC = 10), what is the relationship between (DE) and (BC)?
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In triangle ( riangle ABC), points (D) and (E) lie on sides (AB) and (AC) respectively, such that (DE \parallel BC). If (AD = 2), (DB = 4), and (AE = 3), what is the length of (EC)?
In triangle ( riangle ABC), points (D) and (E) lie on sides (AB) and (AC) respectively, such that (DE \parallel BC). If (AD = 2), (DB = 4), and (AE = 3), what is the length of (EC)?
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Two triangles, ( riangle ABC) and ( riangle DEF), have equal heights. If the base of ( riangle ABC) is three times the base of ( riangle DEF), what is the ratio of the area of ( riangle ABC) to the area of ( riangle DEF)?
Two triangles, ( riangle ABC) and ( riangle DEF), have equal heights. If the base of ( riangle ABC) is three times the base of ( riangle DEF), what is the ratio of the area of ( riangle ABC) to the area of ( riangle DEF)?
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Two triangles, ( riangle ABC) and ( riangle DEF), are similar. If the perimeter of ( riangle ABC) is 12 cm and the perimeter of ( riangle DEF) is 24 cm, what is the ratio of the area of ( riangle ABC) to the area of ( riangle DEF)?
Two triangles, ( riangle ABC) and ( riangle DEF), are similar. If the perimeter of ( riangle ABC) is 12 cm and the perimeter of ( riangle DEF) is 24 cm, what is the ratio of the area of ( riangle ABC) to the area of ( riangle DEF)?
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Given that (\cos(\alpha - \beta) = \frac{1}{2}) and (\sin(\alpha - \beta) = \frac{\sqrt{3}}{2}), what is the value of (\tan(\alpha - \beta))?
Given that (\cos(\alpha - \beta) = \frac{1}{2}) and (\sin(\alpha - \beta) = \frac{\sqrt{3}}{2}), what is the value of (\tan(\alpha - \beta))?
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Which of the following expressions is equivalent to (\sin(\alpha + \beta)) in terms of (\cos(\alpha - \beta)) and the even-odd identities?
Which of the following expressions is equivalent to (\sin(\alpha + \beta)) in terms of (\cos(\alpha - \beta)) and the even-odd identities?
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Which of the following expressions is equivalent to (\cos(\alpha + \beta)) using only the cosine difference formula and the even-odd identities?
Which of the following expressions is equivalent to (\cos(\alpha + \beta)) using only the cosine difference formula and the even-odd identities?
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In deriving the cosine difference formula, we equate the square of the distance between points K and L on the unit circle, derived using the distance formula and the cosine rule. What is the expression for the square of the distance between points K and L, using the cosine rule?
In deriving the cosine difference formula, we equate the square of the distance between points K and L on the unit circle, derived using the distance formula and the cosine rule. What is the expression for the square of the distance between points K and L, using the cosine rule?
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Using the compound angle identities, find the exact value of (\sin 15°) in terms of radicals.
Using the compound angle identities, find the exact value of (\sin 15°) in terms of radicals.
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The cosine difference formula can be used to prove the cosine sum formula by rewriting the angle (\alpha + \beta)) as a difference. What is this equivalent difference?
The cosine difference formula can be used to prove the cosine sum formula by rewriting the angle (\alpha + \beta)) as a difference. What is this equivalent difference?
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Given that (\cos \alpha = \frac{3}{5}) and (\sin \beta = \frac{12}{13}), where (0° < \alpha < 90°) and (90° < \beta < 180°), what is the value of (\cos(\alpha + \beta))?
Given that (\cos \alpha = \frac{3}{5}) and (\sin \beta = \frac{12}{13}), where (0° < \alpha < 90°) and (90° < \beta < 180°), what is the value of (\cos(\alpha + \beta))?
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Which of the following is NOT a valid compound angle identity?
Which of the following is NOT a valid compound angle identity?
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