Gr12 Mathematics: June Medium P(2)
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Questions and Answers

What is the formula for the area of a triangle?

  • Base × Height
  • Base - Height
  • 0.5 × Base × Height (correct)
  • Base + Height

    • What is the formula for the area of a rhombus?

  • Diagonal AC - Diagonal BD
  • 0.5 × Diagonal AC × Diagonal BD (correct)
  • Diagonal AC + Diagonal BD
  • Diagonal AC × Diagonal BD

    • What is the formula for the area of a trapezium?

  • Base1 + Base2 + Height
  • Base1 - Base2 - Height
  • 0.5 × (Base1 + Base2) × Height (correct)
  • Base1 × Base2 × Height

    • What is the Basic Proportionality Theorem also known as?

    <p>Thales' Theorem</p> Signup and view all the answers

    What is the condition for similarity of two triangles?

    <p>Equal corresponding angles</p> Signup and view all the answers

    What is the conclusion about triangles with the same height?

    <p>They have proportional areas</p> Signup and view all the answers

    What is the formula for the area of a square?

    <p>Side^2</p> Signup and view all the answers

    What is the definition of a polygon?

    <p>A plane, closed shape consisting of three or more line segments</p> Signup and view all the answers

    What is the formula for the area of a parallelogram?

    <p>Base × Height</p> Signup and view all the answers

    What is the conclusion about triangles with the same base?

    <p>They have equal areas</p> Signup and view all the answers

    What is the equation of a circle with center at ( (2, -3) ) and radius 5?

    <p>( (x - 2)^2 + (y + 3)^2 = 25 )</p> Signup and view all the answers

    What is the center and radius of the circle with the equation ( x^2 + y^2 - 6x + 4y - 12 = 0 )?

    <p>Center: ( (3, -2) ), radius: 5</p> Signup and view all the answers

    Which of the following is a correct statement about the symmetry of a circle with center at the origin?

    <p>The circle is symmetric about both the x-axis and the y-axis.</p> Signup and view all the answers

    Which of the following equations represents a circle with center at ( (0, 0) ) and radius 4?

    <p>( x^2 + y^2 = 16 )</p> Signup and view all the answers

    What is the radius of the circle defined by the equation ( x^2 + y^2 + 8x - 10y + 16 = 0 )?

    <p>5</p> Signup and view all the answers

    What is the equation of a circle with a center at ( (-1, 3) ) and a diameter of 10?

    <p>( (x + 1)^2 + (y - 3)^2 = 25 )</p> Signup and view all the answers

    What is the center of the circle with the equation ( (x + 5)^2 + (y - 2)^2 = 9 )?

    <p>( (-5, 2) )</p> Signup and view all the answers

    Which of the following points lies on the circle with the equation ( x^2 + y^2 = 25 )?

    <p>All of the above</p> Signup and view all the answers

    If a line segment divides two sides of a triangle proportionally, what can we conclude about the line segment?

    <p>The line segment is parallel to the third side of the triangle.</p> Signup and view all the answers

    Two triangles are similar if they have the same shape but differ in size. Which of the following conditions must be met for two triangles to be similar?

    <p>Both a and c.</p> Signup and view all the answers

    A line segment is drawn connecting the midpoints of two sides of a triangle. What can we conclude about this line segment?

    <p>The line segment is parallel to the third side of the triangle and half its length.</p> Signup and view all the answers

    Two triangles are on the same side of the same base and have equal areas. What can we conclude about the triangles?

    <p>The triangles lie between parallel lines.</p> Signup and view all the answers

    Given two triangles, ( riangle ABC) and ( riangle DEF), where (\angle A = \angle D), (\angle B = \angle E), and (\angle C = \angle F). Which of the following statements is TRUE?

    <p>The triangles are similar.</p> Signup and view all the answers

    If a line is drawn parallel to one side of a triangle and divides the other two sides proportionally, what is the ratio of the areas of the two triangles created?

    <p>The ratio of the areas is equal to the square of the ratio of the corresponding sides.</p> Signup and view all the answers

    A triangle has sides of length 6 cm, 8 cm, and 10 cm. Another triangle has sides of length 3 cm, 4 cm, and 5 cm. Are these triangles similar?

    <p>Yes, because the sides are in the same proportion.</p> Signup and view all the answers

    In ( riangle ABC), point D is the midpoint of side AB and point E is the midpoint of side AC. What can we conclude about DE?

    <p>DE is parallel to BC and DE = (rac{1}{2})BC</p> Signup and view all the answers

    Two triangles have equal bases and lie between the same parallel lines. What can we conclude about their areas?

    <p>The areas are equal.</p> Signup and view all the answers

    If a line segment divides two sides of a triangle proportionally, and the line segment is not parallel to the third side, what can we conclude about the triangle?

    <p>We cannot conclude anything specific about the triangle.</p> Signup and view all the answers

    If tan θ = x, what is the general solution for θ?

    <p>θ = tan⁻¹ x + k · 180°</p> Signup and view all the answers

    What is the formula to find the area of a triangle ABC?

    <p>Area △ABC = (1/2)bc sin A</p> Signup and view all the answers

    When should you use the Sine Rule?

    <p>When no right angle is given, and two sides and an angle (not the included angle) are known</p> Signup and view all the answers

    What is the formula to find the height of a pole?

    <p>h = (d sin α) / (tan β)</p> Signup and view all the answers

    What is the first step in solving three-dimensional problems?

    <p>Draw a sketch to visualize the problem</p> Signup and view all the answers

    What is the formula to find the height of a building?

    <p>h = (b sin α) / (sin (β + θ))</p> Signup and view all the answers

    When should you use the Cosine Rule?

    <p>When no right angle is given, and three sides are known</p> Signup and view all the answers

    What is the application of trigonometric functions in real-life problems?

    <p>To find the height of a pole or building</p> Signup and view all the answers

    In the proof of the Pythagorean theorem, why are triangles ABD and CBA similar?

    <p>They have the same three angles.</p> Signup and view all the answers

    Which of the following statements is NOT a direct consequence of the proportionality theorem?

    <p>The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.</p> Signup and view all the answers

    In the proof of the Pythagorean theorem, how is the relationship between (AB^2) and (BD \cdot BC) established?

    <p>By the similarity of triangles ABD and CBA.</p> Signup and view all the answers

    What is the main reason why (GH \parallel BC) in the proof that triangles with sides in proportion are similar?

    <p>The sides are in proportion.</p> Signup and view all the answers

    In the proof that triangles with sides in proportion are similar, why is (GH = EF)?

    <p>They are constructed to be equal.</p> Signup and view all the answers

    What does the converse of the Pythagorean theorem state?

    <p>If the square on the hypotenuse is equal to the sum of the squares on the other two sides, then the triangle is a right-angled triangle.</p> Signup and view all the answers

    What is the purpose of drawing line segment AD perpendicular to BC in the proof of the Pythagorean theorem?

    <p>To create similar triangles ABD and CAD.</p> Signup and view all the answers

    Which of the following statements is NOT a valid application of the proportionality theorem?

    <p>If two triangles have the same base, then their areas are proportional to their heights.</p> Signup and view all the answers

    What is the relationship between the areas of two triangles with equal heights?

    <p>The areas are proportional to their bases.</p> Signup and view all the answers

    If two triangles are equiangular, what can you conclude about their corresponding sides?

    <p>They are in proportion.</p> Signup and view all the answers

    What condition must be satisfied for two polygons to be similar?

    <p>All corresponding angles are equal, and all corresponding sides are in the same proportion.</p> Signup and view all the answers

    Which formula correctly represents the sine of a sum of two angles?

    <p>$ ext{sin}(eta + heta) = ext{sin} eta ext{cos} heta + ext{cos} eta ext{sin} heta $</p> Signup and view all the answers

    What is the correct formulation of the cosine of a difference of two angles?

    <p>$ ext{cos}(eta - heta) = ext{cos} eta ext{cos} heta + ext{sin} eta ext{sin} heta $</p> Signup and view all the answers

    Which identity is used to derive the cosine of a sum of two angles?

    <p>Negative angle identity</p> Signup and view all the answers

    What is the correct expression for $ ext{sin}(eta - heta)$?

    <p>$ ext{sin}(eta - heta) = ext{sin} eta ext{cos} heta - ext{cos} eta ext{sin} heta $</p> Signup and view all the answers

    What must be established to prove similarity in triangles?

    <p>Equiangularity or proportionality of sides</p> Signup and view all the answers

    In deriving the formula for $ ext{cos}(eta + heta)$, which identities are applied?

    <p>Even-odd identities</p> Signup and view all the answers

    Which of the following is true for similar triangles?

    <p>Corresponding sides are in proportion and all angles are equal.</p> Signup and view all the answers

    What is the sine of the angle difference represented by the formula $\sin(\alpha - \beta)$?

    <p>$\sin \alpha \cos \beta - \cos \alpha \sin \beta$</p> Signup and view all the answers

    How can $\cos(90^ ext{o} - \alpha)$ be expressed using a co-function identity?

    <p>$\sin \alpha$</p> Signup and view all the answers

    What is the cosine of the sum of two angles $\cos(\alpha + \beta)$?

    <p>$\cos \alpha \cos \beta - \sin \alpha \sin \beta$</p> Signup and view all the answers

    What is the formula for the sine of the double angle $\sin(2\alpha)$?

    <p>$2 \sin \alpha \cos \alpha$</p> Signup and view all the answers

    Which of the following represents one of the forms of the cosine of double angle $\cos(2\alpha)$?

    <p>$2 \cos^2 \alpha - 1$</p> Signup and view all the answers

    Which step is NOT a part of the general solution method for solving trigonometric equations?

    <p>Draw the Cartesian coordinates</p> Signup and view all the answers

    What is the relationship between the sine and cosine functions at complementary angles?

    <p>Both a and b</p> Signup and view all the answers

    Which formula represents the cosine of a difference $\cos(\alpha - \beta)$?

    <p>$\cos \alpha \cos \beta + \sin \alpha \sin \beta$</p> Signup and view all the answers

    What is the first step in finding a general solution to a trigonometric equation?

    <p>Simplify the equation</p> Signup and view all the answers

    What is the value of the radius, $r$, in terms of $D$, $E$, and $F$ from the given standard form of the circle?

    <p>$ ext{sqrt}igg{(igg(rac{D}{2}igg)^2 + igg(rac{E}{2}igg)^2 - Figg{)}$</p> Signup and view all the answers

    What relationship holds between the gradients of the radius and the tangent at the point of tangency on the circle?

    <p>$m_{radius} imes m_{tangent} = -1$</p> Signup and view all the answers

    Which equation represents the gradient of the radius from the center of the circle to the point of tangency?

    <p>$m_{radius} = rac{y_1 - b}{x_1 - a}$</p> Signup and view all the answers

    What form should the equation of the tangent line take?

    <p>$y - y_1 = m_{tangent}(x - x_1)$</p> Signup and view all the answers

    What is a defining characteristic of a tangent line in relation to a circle?

    <p>It intersects the circle at exactly one point.</p> Signup and view all the answers

    According to the properties of proportion, what does cross multiplication of the ratios $rac{w}{x} = rac{y}{z}$ yield?

    <p>$w imes z = x imes y$</p> Signup and view all the answers

    What is a key feature of ratios?

    <p>Ratios are unit-less comparisons of similar quantities.</p> Signup and view all the answers

    If a line segment divides two sides of a triangle proportionally, which theorem is applied?

    <p>Basic Proportionality Theorem (Thales' theorem)</p> Signup and view all the answers

    Which of the following is NOT a property of proportion?

    <p>Distributive Proportion</p> Signup and view all the answers

    To solve a proportional problem, what is the first step?

    <p>Identify the given ratios.</p> Signup and view all the answers

    What is the equation of a circle with center at (0, 0) and radius r?

    <p>x^2 + y^2 = r^2</p> Signup and view all the answers

    What is the equation of a circle with center at (a, b) and radius r?

    <p>(x - a)^2 + (y - b)^2 = r^2</p> Signup and view all the answers

    What is the method used to find the center and radius of a circle given its general equation?

    <p>Completing the square</p> Signup and view all the answers

    What is the symmetry of a circle with center at the origin?

    <p>About the x-axis, y-axis, origin, and lines y = x and y = -x</p> Signup and view all the answers

    What is the first step in finding the equation of a circle with center at (a, b) and radius r?

    <p>Consider a point P(x, y) on the circumference of the circle</p> Signup and view all the answers

    What is the purpose of squaring both sides when deriving the equation of a circle?

    <p>To eliminate the square root</p> Signup and view all the answers

    What is the significance of the point P(x, y) in deriving the equation of a circle?

    <p>It is a point on the circumference of the circle</p> Signup and view all the answers

    What is the advantage of rewriting a circle's equation in standard form?

    <p>It allows us to easily find the center and radius of the circle</p> Signup and view all the answers

    What is the relationship between the gradient of the radius and the gradient of the tangent at the point of tangency?

    <p>m_radius × m_tangent = -1</p> Signup and view all the answers

    What is the formula for the equation of a tangent to a circle at the point of tangency (x₁, y₁)?

    <p>y - y₁ = m (x - x₁)</p> Signup and view all the answers

    What is the property of proportion that states wz = xy?

    <p>Cross Multiplication</p> Signup and view all the answers

    What is the condition for similarity of two triangles?

    <p>The corresponding sides are proportional</p> Signup and view all the answers

    What is the name of the theorem that states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally?

    <p>Thales' Theorem</p> Signup and view all the answers

    What is the ratio of the areas of two triangles with equal heights?

    <p>The ratio of their bases</p> Signup and view all the answers

    What is the condition for two ratios to be in proportion?

    <p>The ratios are equal</p> Signup and view all the answers

    What is the name of the concept that compares two quantities with the same units?

    <p>Ratio</p> Signup and view all the answers

    What is the application of proportion in geometry?

    <p>Comparing sides of triangles</p> Signup and view all the answers

    What is the equation of a circle in standard form?

    <p>(x - a)² + (y - b)² = r²</p> Signup and view all the answers

    What is the formula to find the area of a triangle using the sine function?

    <p>$\frac{1}{2}bc \sin A$</p> Signup and view all the answers

    When should you use the Cosine Rule?

    <p>When no right angle is given, and either two sides and the included angle or three sides are given.</p> Signup and view all the answers

    What is the formula to find the height of a pole?

    <p>$h = \frac{d \sin \alpha}{\sin \beta} \tan \beta$</p> Signup and view all the answers

    What is the first step in solving three-dimensional problems?

    <p>Draw a sketch of the problem.</p> Signup and view all the answers

    What is the formula to find the height of a building?

    <p>$h = \frac{b \sin \alpha \sin \theta}{\sin(\beta + \theta)}$</p> Signup and view all the answers

    When should you use the Sine Rule?

    <p>When no right angle is given, and two sides and an angle (not the included angle) are given.</p> Signup and view all the answers

    What is the application of trigonometric functions in real-life problems?

    <p>All of the above.</p> Signup and view all the answers

    What is the general solution for θ when tan θ = x?

    <p>$θ = \tan^{-1} x + k \cdot 360^\circ$</p> Signup and view all the answers

    In a triangle (\triangle ABC), a line segment (DE) is drawn parallel to side (BC), with (D) on (AB) and (E) on (AC). If (AD = 4) and (DB = 6), what is the length of (AE) if (EC = 9)?

    <p>12</p> Signup and view all the answers

    Two triangles, (\triangle ABC) and (\triangle DEF), are equiangular. If (AB = 8), (BC = 12), and (DE = 6), what is the length of (EF)?

    <p>9</p> Signup and view all the answers

    If two triangles have the same height, what is the ratio of their areas?

    <p>Equal to the ratio of their bases</p> Signup and view all the answers

    If two triangles have the same base and lie between the same parallel lines, what is the relationship between their areas?

    <p>The areas are equal</p> Signup and view all the answers

    In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. This theorem is known as:

    <p>Pythagoras' Theorem</p> Signup and view all the answers

    Which of the following is NOT a property of similar polygons?

    <p>The polygons have the same area</p> Signup and view all the answers

    In a triangle (\triangle ABC), point (D) is the midpoint of side (AB), and point (E) is the midpoint of side (AC). What is the relationship between line segment (DE) and side (BC)?

    <p>DE is parallel to BC and half the length of BC</p> Signup and view all the answers

    Two triangles have the same base and lie between the same parallel lines. What is the ratio of their areas?

    <p>The ratio of their areas is 1:1</p> Signup and view all the answers

    In (\triangle ABC), a line segment (DE) is drawn parallel to side (BC), with (D) on (AB) and (E) on (AC). If (AD = 3), (DB = 5), and (AE = 6), what is the length of (EC)?

    <p>10</p> Signup and view all the answers

    Two triangles are similar if and only if:

    <p>Their corresponding angles are equal</p> Signup and view all the answers

    Using the cosine difference formula, what is the correct expression for (cos (\alpha - \beta))?

    <p>(cos \alpha cos \beta + sin \alpha sin \beta)</p> Signup and view all the answers

    Which of the following is NOT a correct compound angle identity?

    <p>(sin(\alpha - \beta) = sin \alpha cos \beta + cos \alpha sin \beta)</p> Signup and view all the answers

    When deriving the formula for (cos(\alpha + \beta)), which identity is used to rewrite the angle as a difference?

    <p>Negative angle identity</p> Signup and view all the answers

    What is the correct expression for (cos (\alpha + \beta)), derived using the negative angle identity?

    <p>(cos \alpha cos \beta - sin \alpha sin \beta)</p> Signup and view all the answers

    What is the primary method used to derive the cosine difference formula, (cos(\alpha - \beta))?

    <p>Using the distance formula and the cosine rule</p> Signup and view all the answers

    For which of the following expressions are we able to directly apply a compound angle identity?

    <p>(cos(\alpha - \beta))</p> Signup and view all the answers

    Which of the following is a valid application of the compound angle identity for the cosine of a difference, (cos(\alpha - \beta) = cos \alpha cos \beta + sin \alpha sin \beta)?

    <p>(cos(60° - 30°) = cos 60° cos 30° + sin 60° sin 30°)</p> Signup and view all the answers

    Which of the following statements accurately describes the application of compound angle identities in trigonometry?

    <p>All of the above statements are correct.</p> Signup and view all the answers

    What can be concluded if two triangles have corresponding sides that are in proportion?

    <p>The triangles are similar.</p> Signup and view all the answers

    In the proof of the Pythagorean theorem, what construction helps establish that triangles ABD and CBA are similar?

    <p>Drawing line AD perpendicular to BC.</p> Signup and view all the answers

    Which statement correctly represents the relationship of angles in similar triangles?

    <p>Corresponding angles are equal.</p> Signup and view all the answers

    If two triangles are equiangular, what can be said about their sides?

    <p>Their corresponding sides are in proportion.</p> Signup and view all the answers

    What must be proven to confirm that triangles ABC and DEF are similar?

    <p>The corresponding angles are equal.</p> Signup and view all the answers

    What does the Pythagorean theorem demonstrate about the sides of a right triangle?

    <p>The square of the hypotenuse equals the sum of the squares of the other two sides.</p> Signup and view all the answers

    In the context of similarity, which of the following is NOT a requirement for two triangles to be classified as similar?

    <p>The triangles are drawn to scale.</p> Signup and view all the answers

    What is the converse of the Pythagorean theorem?

    <p>If the square of one side equals the sum of the squares of the other two, the triangle is right-angled.</p> Signup and view all the answers

    Which of the following is a correct expression for (\cos(2\alpha)) based on the double angle formula?

    <p>(2\cos^2 \alpha - 1)</p> Signup and view all the answers

    Which statement accurately relates the areas of triangles with equal bases and heights?

    <p>The areas are proportional to their heights.</p> Signup and view all the answers

    What is the correct formula for the sine of a double angle?

    <p>(\sin(2\alpha) = 2\sin \alpha \cos \alpha)</p> Signup and view all the answers

    Which of the following steps is NOT involved in finding the general solution of a trigonometric equation?

    <p>Calculate the inverse tangent of both sides of the equation.</p> Signup and view all the answers

    What is the correct formula for the cosine of a difference of two angles?

    <p>(\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta)</p> Signup and view all the answers

    Which trigonometric identity is used to derive the double angle formula for cosine in the form (\cos(2\alpha) = 2\cos^2 \alpha - 1)?

    <p>(\sin^2 \alpha + \cos^2 \alpha = 1)</p> Signup and view all the answers

    Which of the following is a correct expression for (\sin(\alpha + \beta)) based on the compound angle formulas?

    <p>(\sin \alpha \cos \beta + \cos \alpha \sin \beta)</p> Signup and view all the answers

    Which of the following steps is NOT involved in the derivation of the double angle formula for sine?

    <p>Use the Pythagorean identity to simplify the expression.</p> Signup and view all the answers

    How many solutions does the equation (\sin x = 1/2) have in the interval ([0, 360^\circ]) ?

    <p>Two</p> Signup and view all the answers

    What is the general solution for the equation (\cos x = -1)?

    <p>(x = 180^\circ + 360^\circ n, n \in Z)</p> Signup and view all the answers

    Which of the following statements is TRUE regarding the derivation of the double angle formula for cosine?

    <p>The formula is derived using both the sum formula for cosine and the Pythagorean identity.</p> Signup and view all the answers

    In a triangle, a line is drawn parallel to one side of the triangle, dividing the other two sides proportionally. What can we conclude about the line drawn?

    <p>The line divides the triangle into two similar triangles.</p> Signup and view all the answers

    If two triangles have the same base and equal areas, what can we conclude about their heights?

    <p>The heights are equal.</p> Signup and view all the answers

    Given two triangles, (\triangle ABC) and (\triangle DEF), where (\angle A = \angle D), (\angle B = \angle E), and (\angle C = \angle F), what can we conclude about the triangles?

    <p>The triangles are similar.</p> Signup and view all the answers

    A line segment connects the midpoints of two sides of a triangle. What is true about the line segment?

    <p>It is parallel to the third side and half its length.</p> Signup and view all the answers

    Which of the following is NOT a necessary condition for two polygons to be similar?

    <p>All pairs of corresponding sides are equal.</p> Signup and view all the answers

    Two triangles have equal bases and lie between the same parallel lines. Which of the following statements is true about their areas?

    <p>Their areas are equal.</p> Signup and view all the answers

    If two triangles are equiangular, what is the relationship between their corresponding sides?

    <p>The corresponding sides are proportional.</p> Signup and view all the answers

    A line is drawn parallel to one side of a triangle, dividing the other two sides proportionally. What is the ratio of the areas of the two triangles created?

    <p>The ratio of the areas is equal to the square of the ratio of the corresponding sides.</p> Signup and view all the answers

    If a line segment divides two sides of a triangle proportionally, and the line segment is not parallel to the third side, what can we conclude about the triangle?

    <p>We cannot conclude anything specific about the triangle.</p> Signup and view all the answers

    What is the main reason why (GH \parallel BC) in the proof that triangles with sides in proportion are similar?

    <p>Because (\angle AGH = \angle B) and (\angle AHG = \angle C).</p> Signup and view all the answers

    Two triangles have the same base and lie between the same parallel lines. What can we conclude about their areas?

    <p>The areas are equal.</p> Signup and view all the answers

    If a line segment divides two sides of a triangle proportionally, what can we conclude about the line segment?

    <p>The line segment is parallel to the third side.</p> Signup and view all the answers

    In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. This is known as:

    <p>Pythagoras' Theorem</p> Signup and view all the answers

    Two triangles are equiangular, meaning they have the same angles. What can we conclude about their corresponding sides?

    <p>The corresponding sides are proportional.</p> Signup and view all the answers

    What is the ratio of the areas of two triangles with equal heights?

    <p>The ratio is equal to the ratio of their bases.</p> Signup and view all the answers

    Which of the following is NOT a valid application of the proportionality theorem?

    <p>Calculating the area of a triangle.</p> Signup and view all the answers

    If a line segment divides two sides of a triangle proportionally, and the line segment is not parallel to the third side, what can we conclude about the triangle?

    <p>We cannot conclude anything about the triangle.</p> Signup and view all the answers

    Two triangles have equal bases and lie between the same parallel lines. What can we conclude about their areas?

    <p>The areas are equal.</p> Signup and view all the answers

    In the proof of the Pythagorean theorem, why is the relationship between (AB^2) and (BD \cdot BC) established?

    <p>Using the similarity of triangles ABD and CBA.</p> Signup and view all the answers

    What is the equation of a circle with center at (2, -3) and radius 5?

    <p>(x - 2)^2 + (y + 3)^2 = 25</p> Signup and view all the answers

    What is the center and radius of the circle with the equation x^2 + y^2 - 6x + 4y - 12 = 0?

    <p>Center: (3, -2), Radius: 5</p> Signup and view all the answers

    Which of the following equations represents a circle with center at (0, 0) and radius 4?

    <p>x^2 + y^2 = 16</p> Signup and view all the answers

    What is the radius of the circle defined by the equation x^2 + y^2 + 8x - 10y + 16 = 0?

    <p>5</p> Signup and view all the answers

    Which of the following points lies on the circle with the equation x^2 + y^2 = 25?

    <p>(4, 3)</p> Signup and view all the answers

    What is the equation of a circle with a center at (-1, 3) and a diameter of 10?

    <p>(x + 1)^2 + (y - 3)^2 = 25</p> Signup and view all the answers

    What is the center of the circle with the equation (x + 5)^2 + (y - 2)^2 = 9?

    <p>(-5, 2)</p> Signup and view all the answers

    What is the equation of a circle with center at (0, 0) and radius 3?

    <p>x^2 + y^2 = 9</p> Signup and view all the answers

    What is the formula for the cosine of a difference of two angles?

    <p>cos(α - β) = cosαcosβ + sinαsinβ</p> Signup and view all the answers

    What method is used to derive the formula for cos(α - β)?

    <p>Using the distance formula and cosine rule</p> Signup and view all the answers

    What is the formula for the sine of a sum of two angles?

    <p>sin(α + β) = sinαcosβ + cosαsinβ</p> Signup and view all the answers

    What is the condition for similarity of polygons?

    <p>All pairs of corresponding angles are equal, and all pairs of corresponding sides are in the same proportion</p> Signup and view all the answers

    What is the formula for the cosine of a sum of two angles?

    <p>cos(α + β) = cosαcosβ - sinαsinβ</p> Signup and view all the answers

    What method is used to derive the formula for cos(α + β)?

    <p>Using the negative angle identity</p> Signup and view all the answers

    What is the formula for the sine of a difference of two angles?

    <p>sin(α - β) = sinαcosβ - cosαsinβ</p> Signup and view all the answers

    What is the condition for similarity of triangles?

    <p>Prove either equiangularity or proportionality of sides</p> Signup and view all the answers

    What is the formula for the sine of a difference of two angles?

    <p>sin(α - β) = sin α cos β - cos α sin β</p> Signup and view all the answers

    What is the formula for the cosine of a sum of two angles?

    <p>cos(α + β) = cos α cos β - sin α sin β</p> Signup and view all the answers

    What is the formula for the sine of a double angle?

    <p>sin(2α) = 2 sin α cos α</p> Signup and view all the answers

    What is the formula for the cosine of a double angle?

    <p>cos(2α) = 2 cos^2 α - 1</p> Signup and view all the answers

    What is the purpose of using a CAST diagram in solving trigonometric equations?

    <p>To determine where the function is positive or negative</p> Signup and view all the answers

    What is the general solution method for solving trigonometric equations?

    <p>Simplify, find the reference angle, and use CAST diagram</p> Signup and view all the answers

    What is the purpose of finding the reference angle in solving trigonometric equations?

    <p>To determine the restricted values within a specified interval</p> Signup and view all the answers

    What is the formula for the cosine of a difference of two angles?

    <p>cos(α - β) = cos α cos β - sin α sin β</p> Signup and view all the answers

    What is the difference between the sine of a sum and the sine of a difference of two angles?

    <p>The sign of the second term is changed</p> Signup and view all the answers

    Why are double angle formulas useful in trigonometry?

    <p>They can be used to relate the trigonometric functions of an angle to those of its double angle</p> Signup and view all the answers

    What is the general solution for θ if tan θ = x?

    <p>θ = tan⁻¹ x + k ⋅ 180˚ or θ = 180˚ - tan⁻¹ x + k ⋅ 180˚</p> Signup and view all the answers

    What is the first step in solving three-dimensional problems?

    <p>Draw a sketch</p> Signup and view all the answers

    When should you use the Sine Rule?

    <p>When no right angle is given, and either two sides and an angle (not the included angle) or two angles and a side are given.</p> Signup and view all the answers

    What is the formula to find the height of a pole?

    <p>h = d sin α / sin β</p> Signup and view all the answers

    What is the formula to find the area of a triangle ABC?

    <p>Area = (1/2)ab sin C</p> Signup and view all the answers

    When should you use the Cosine Rule?

    <p>When no right angle is given, and either two sides and the included angle or three sides are given.</p> Signup and view all the answers

    What is the formula to find the height of a building?

    <p>h = b sin α sin θ / sin(β + θ)</p> Signup and view all the answers

    What is the application of trigonometric functions in real-life problems?

    <p>In calculating distances, heights, and angles in various fields such as physics, engineering, and navigation</p> Signup and view all the answers

    From the standard form of a circle equation, identify the center coordinates.

    <p>(-D/2, -E/2)</p> Signup and view all the answers

    What is the relationship between the gradient of the radius and the gradient of the tangent?

    <p>They multiply to equal -1.</p> Signup and view all the answers

    To determine the equation of a tangent at point (x1, y1), which form is used?

    <p>y - y1 = m_tangent (x - x1)</p> Signup and view all the answers

    In a proportional relationship, what does cross multiplication state?

    <p>w * z = x * y</p> Signup and view all the answers

    When is a proportion considered valid in terms of triangle sides?

    <p>If two ratios are equal.</p> Signup and view all the answers

    What is the definition of a tangent line to a circle?

    <p>A line that touches the circle at exactly one point.</p> Signup and view all the answers

    How can ratios be expressed?

    <p>In forms such as fractions, colon format, or in words.</p> Signup and view all the answers

    What does the Basic Proportionality Theorem state about parallel lines in triangles?

    <p>It shows that equal ratios exist between divided sides.</p> Signup and view all the answers

    What is the formula to find the radius 'r' of a circle from its standard form equation?

    <p>r = \sqrt{(D/2)^2 + (E/2)^2 - F}</p> Signup and view all the answers

    What is a key property of ratios?

    <p>They should be simplified to their simplest form.</p> Signup and view all the answers

    If the sides of triangles ABC and DEF are in proportion, what conclusion can be drawn about their angles?

    <p>The angles of the triangles are equal.</p> Signup and view all the answers

    In the proof of the Pythagorean theorem, what does the construction of line segment AD being perpendicular to BC establish?

    <p>It confirms that triangle ABD is a right triangle.</p> Signup and view all the answers

    What is demonstrated by the statement \(rac{AB}{DE} = rac{AC}{DF} = rac{BC}{EF}\)?

    <p>These triangles ABC and DEF are similar.</p> Signup and view all the answers

    What is the significance of drawing line segment GH parallel to BC in the proof that triangles with proportional sides are similar?

    <p>It creates corresponding angles that are equal.</p> Signup and view all the answers

    In proving the Pythagorean theorem, how are the relationships derived from the similarity of triangles ABD and CBA used?

    <p>To establish ratios between all sides.</p> Signup and view all the answers

    Which statement is true regarding the areas of two triangles with equal bases lying between the same parallel lines?

    <p>They will have equal areas.</p> Signup and view all the answers

    Which theorem states that if the square of one side of a triangle is equal to the sum of the squares of the other two sides, then it contains a right angle?

    <p>The Converse of the Pythagorean Theorem.</p> Signup and view all the answers

    According to the proportionality in triangles, how can areas be calculated if two triangles have the same height?

    <p>The area will be proportional to their bases.</p> Signup and view all the answers

    In the context of similar triangles, what is necessary to conclude that two triangles are similar?

    <p>Their corresponding sides must be in the same ratio.</p> Signup and view all the answers

    Which is NOT a characteristic of equiangular triangles?

    <p>They have equal areas regardless of sides.</p> Signup and view all the answers

    If a line drawn parallel to one side of a triangle divides the other two sides proportionally, what is the name of this theorem?

    <p>Proportionality Theorem</p> Signup and view all the answers

    Two triangles are similar if they have the same shape but differ in size. Which of the following conditions must be met for two triangles to be similar?

    <p>All corresponding angles are equal and all corresponding sides are in the same proportion.</p> Signup and view all the answers

    Which of the following statements is TRUE about triangles on the same base and equal in area?

    <p>They must lie between parallel lines.</p> Signup and view all the answers

    In the proof that triangles with sides in proportion are similar, why is (GH = EF)?

    <p>Because (GH \parallel BC) and (\triangle ABC) and (\triangle DEF) are similar.</p> Signup and view all the answers

    Which of the following statements is NOT a valid application of the proportionality theorem?

    <p>Calculating the area of a triangle when a line parallel to one side divides the other two sides proportionally.</p> Signup and view all the answers

    A line segment is drawn connecting the midpoints of two sides of a triangle. What can we conclude about this line segment?

    <p>It is parallel to the third side and half its length.</p> Signup and view all the answers

    Which of the following statements is NOT a direct consequence of the proportionality theorem?

    <p>A line drawn parallel to one side of a triangle divides the other two sides proportionally.</p> Signup and view all the answers

    What is the main reason why (GH \parallel BC) in the proof that triangles with sides in proportion are similar?

    <p>Because corresponding angles are equal, specifically (\angle AGH = \angle B).</p> Signup and view all the answers

    Given two triangles, (\triangle ABC) and (\triangle DEF), where (\angle A = \angle D), (\angle B = \angle E), and (\angle C = \angle F). Which of the following statements is TRUE?

    <p>The triangles are similar.</p> Signup and view all the answers

    In the proof of the Pythagorean theorem, how is the relationship between (AB^2) and (BD \cdot BC) established?

    <p>By using the fact that triangles ABD and CBA are similar.</p> Signup and view all the answers

    What is the equation of a circle with center at (a, b) and radius r?

    <p>(x - a)^2 + (y - b)^2 = r^2</p> Signup and view all the answers

    What is the symmetry of a circle with center at the origin?

    <p>About the x-axis, y-axis, origin, and lines y = x and y = -x</p> Signup and view all the answers

    How can we rewrite the general form of a circle's equation to find the center and radius?

    <p>By completing the square</p> Signup and view all the answers

    What is the significance of the Pythagorean theorem in deriving the equation of a circle?

    <p>It is used to derive the equation of the circle from the distance formula</p> Signup and view all the answers

    What is the general form of the equation of a circle?

    <p>x^2 + y^2 + Dx + Ey + F = 0</p> Signup and view all the answers

    What is the process of completing the square used for?

    <p>To find the center and radius of a circle</p> Signup and view all the answers

    What is the benefit of rewriting the general form of a circle's equation in standard form?

    <p>It makes it easier to find the center and radius of the circle</p> Signup and view all the answers

    What is the purpose of the distance formula in deriving the equation of a circle?

    <p>To find the distance between two points</p> Signup and view all the answers

    In a triangle, a line is drawn parallel to one of its sides, intersecting the other two sides. What does this line do to the other two sides?

    <p>Divides them proportionally</p> Signup and view all the answers

    For a triangle with sides of length a, b, and c, and angles A, B, and C opposite those sides, which rule would you use if you are given two angles and the included side?

    <p>Sine Rule</p> Signup and view all the answers

    In a triangle where two sides and the included angle are known, which rule is used to find the length of the third side?

    <p>Cosine Rule</p> Signup and view all the answers

    Which rule is most appropriate for calculating the area of a triangle when only two sides and the included angle are given?

    <p>Area Rule</p> Signup and view all the answers

    When solving three-dimensional problems involving a pole, a building, or other objects, what is the first crucial step?

    <p>Draw a sketch</p> Signup and view all the answers

    You are tasked with finding the height of a building. You know the distance from the building, the angle of elevation to the top of the building, and the angle of depression from the top of the building to a point on the ground. Which rule would you use to find the height?

    <p>Sine Rule</p> Signup and view all the answers

    To find the height of a pole, you know the distance from the base of the pole, the angle of elevation from the ground to the top of the pole, and the angle between the pole and the ground. Which trigonometric ratio would you use directly to calculate the height of the pole?

    <p>Tangent</p> Signup and view all the answers

    If you are given the lengths of all three sides of a triangle, which rule would you use to determine one of the angles?

    <p>Cosine Rule</p> Signup and view all the answers

    In a triangle, two sides and the angle opposite one of those sides are known. What rule can be used to find the other angle opposite the remaining side?

    <p>Sine Rule</p> Signup and view all the answers

    What can be concluded if two triangles share the same base and have equal areas?

    <p>They lie between parallel lines.</p> Signup and view all the answers

    If a line is drawn parallel to one side of triangle ABC and divides the sides proportionally, how can we express this relationship?

    <p>$rac{AD}{DB} = rac{AE}{EC}$</p> Signup and view all the answers

    What does the Mid-point Theorem state about the line connecting the midpoints of two sides of a triangle?

    <p>It is parallel to the third side and half its length.</p> Signup and view all the answers

    For two polygons to be classified as similar, which of the following must occur?

    <p>They must have the same number of sides.</p> Signup and view all the answers

    Under which condition can we conclude that two triangles are similar?

    <p>All pairs of corresponding angles are equal.</p> Signup and view all the answers

    In the Converse of the Mid-point Theorem, what is concluded about the line drawn from a midpoint parallel to a side?

    <p>It divides the third side into two equal segments.</p> Signup and view all the answers

    What is the relationship between the radius and the tangent line at the point of tangency on a circle?

    <p>The radius is perpendicular to the tangent line.</p> Signup and view all the answers

    If the equation of a circle is given as ((x - 3)^2 + (y + 2)^2 = 25), what are the coordinates of the center?

    <p>(3, -2)</p> Signup and view all the answers

    Which of the following statements is TRUE regarding equiangular triangles?

    <p>Their corresponding sides are in proportion.</p> Signup and view all the answers

    What is the formula used to express the area of triangles with the same height and different bases?

    <p>Area is proportional to their bases.</p> Signup and view all the answers

    How would you express the tangent line's slope given the slope of the radius is (m_{radius})?

    <p>(-\frac{1}{m_{radius}})</p> Signup and view all the answers

    Which property of proportion states that (w \cdot z = x \cdot y)?

    <p>Cross Multiplication</p> Signup and view all the answers

    Which of the following best describes similar polygons?

    <p>They are related through scaling.</p> Signup and view all the answers

    If a line segment divides two sides of a triangle proportionally, what can be concluded about that line segment?

    <p>It is parallel to the third side.</p> Signup and view all the answers

    Given the standard form of a circle ((x - a)^2 + (y - b)^2 = r^2), what does (r) represent?

    <p>The radius of the circle.</p> Signup and view all the answers

    Using the cosine difference formula, what is the expression for (cos(\alpha - \beta))?

    <p>(cos \alpha cos \beta + sin \alpha sin \beta)</p> Signup and view all the answers

    In the equation of tangents to a circle, how is the tangent line expressed using the gradient?

    <p>(y - y_1 = m_{tangent} (x - x_1))</p> Signup and view all the answers

    Which trigonometric identity is used to derive the formula for (cos(\alpha + \beta)) from the cosine difference formula?

    <p>Negative angle identity</p> Signup and view all the answers

    Which of the following correctly represents a ratio?

    <p>How many times one quantity contains another in the same units.</p> Signup and view all the answers

    What is the formula for (sin(\alpha + \beta))?

    <p>(sin \alpha cos \beta + cos \alpha sin \beta)</p> Signup and view all the answers

    In the derivation of (cos(\alpha - \beta)) using the distance formula and the cosine rule, what is the expression for KL² obtained from the cosine rule?

    <p>(KL^2 = 2 - 2 cos(\alpha - \beta))</p> Signup and view all the answers

    If two ratios (\frac{w}{x}) and (\frac{y}{z}) are equal, which property can be utilized for solving unknowns?

    <p>Cross Multiplication</p> Signup and view all the answers

    What is the relationship between the formulas for (sin(\alpha - \beta)) and (sin(\alpha + \beta)) in terms of their signs?

    <p>They have opposite signs.</p> Signup and view all the answers

    Which of the following is NOT a valid application of compound angle identities?

    <p>Finding the area of a triangle using the sine rule</p> Signup and view all the answers

    Which of the following best describes the relationship between the formulas for (cos(\alpha + \beta)) and (cos(\alpha - \beta))?

    <p>They differ only in the sign of the second term.</p> Signup and view all the answers

    Which of the following is NOT a compound angle identity?

    <p>(sin^2 \theta + cos^2 \theta = 1)</p> Signup and view all the answers

    What must be proven to demonstrate that two triangles are similar based on the proportionality of their sides?

    <p>That their angles are equal</p> Signup and view all the answers

    In the proof of the Pythagorean theorem, what construction is made to demonstrate the relationship between the sides?

    <p>A segment perpendicular to side BC</p> Signup and view all the answers

    Which of the following statements is derived from the concept of triangle similarity?

    <p>If triangles are similar, their corresponding sides are proportional.</p> Signup and view all the answers

    When applying the theorem of similarity, what can be concluded if two triangles are equilateral?

    <p>Their corresponding angles are equal and sides are in proportion.</p> Signup and view all the answers

    How can one derive the relationship $BC^2 = AB^2 + AC^2$ in the proof of the Pythagorean theorem?

    <p>By proving that the triangles are similar and applying the property of proportional sides.</p> Signup and view all the answers

    What conclusion can be drawn from the proportionality theorem regarding triangles with the same base?

    <p>The areas of the triangles are proportional to their heights.</p> Signup and view all the answers

    Which statement accurately describes the conditions for two triangles being similar?

    <p>All corresponding angles must be equal, and all corresponding sides must be proportional.</p> Signup and view all the answers

    Which relationship holds true when two triangles upon the same segment are drawn with equal angles?

    <p>Their corresponding sides are proportional.</p> Signup and view all the answers

    What is the result of having triangles with equal sides but different orientations?

    <p>They can be similar but not congruent.</p> Signup and view all the answers

    In the converse of the Pythagorean theorem, what condition must be satisfied for the angle between two sides?

    <p>It must be a right angle.</p> Signup and view all the answers

    What is the simplified form of the double angle formula for (\cos(2\alpha)) expressed in terms of (\sin(\alpha))?

    <p>(1 - 2\sin^2(\alpha))</p> Signup and view all the answers

    Using the compound angle formulas, what is the expanded form of (\cos(\alpha - \beta))?

    <p>(\cos\alpha\cos\beta + \sin\alpha\sin\beta)</p> Signup and view all the answers

    Which of the following trigonometric identities is NOT a double angle formula?

    <p>(\tan(2\alpha) = \frac{2\tan\alpha}{1 - \tan^2\alpha})</p> Signup and view all the answers

    What is the first step in deriving the double angle formula for sine, (\sin(2\alpha))?

    <p>Apply the sum formula for sine: (\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta).</p> Signup and view all the answers

    What is the correct formulation of the cosine of a sum of two angles, (\cos(\alpha + \beta))?

    <p>(\cos\alpha\cos\beta - \sin\alpha\sin\beta)</p> Signup and view all the answers

    Using the compound angle formulas, what is the expanded form of (\sin(\alpha + \beta))?

    <p>(\sin\alpha\cos\beta + \cos\alpha\sin\beta)</p> Signup and view all the answers

    Which of the following is NOT a valid double angle formula for (\cos(2\alpha))?

    <p>(1 - \sin^2(\alpha))</p> Signup and view all the answers

    What is the correct formulation of the sine of a difference of two angles, (\sin(\alpha - \beta))?

    <p>(\sin\alpha\cos\beta - \cos\alpha\sin\beta)</p> Signup and view all the answers

    What is the simplified form of the double angle formula for (\cos(2\alpha)) expressed in terms of (\cos(\alpha))?

    <p>(2\cos^2(\alpha) - 1)</p> Signup and view all the answers

    Which of the following trigonometric identities is used to derive the double angle formula for (\cos(2\alpha)) from (\cos^2(\alpha) - \sin^2(\alpha))?

    <p>(\sin^2(\alpha) + \cos^2(\alpha) = 1)</p> Signup and view all the answers

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