Gr12 Mathematics: June Hard P(2)

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)?

7.5 (correct)

9

8

10

What is the equation of a circle with center at the origin and radius r?

x^2 + y^2 = -r^2

x^2 - y^2 = r^2

x^2 - y^2 = -r^2

x^2 + y^2 = r^2 (correct)

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

About the lines y = x and y = -x only

Only about the y-axis

About the x-axis, y-axis, origin, and the lines y = x and y = -x (correct)

Only about the x-axis

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

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

What is the purpose of completing the square in the context of circle equations?

To rewrite the general form of a circle's equation into the standard form Signup and view all the answers

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

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

What is the first step in completing the square to find the center and radius of a circle?

Group the x terms and the y terms Signup and view all the answers

What is the equation derived from the Pythagorean theorem?

x^2 + y^2 = r^2 Signup and view all the answers

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

m_radius × m_tangent = -1 Signup and view all the answers

What is the purpose of using the distance formula in the context of circle equations?

To derive the equation of a circle with center at (a, b) Signup and view all the answers

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

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

What is the key concept in determining the equation of a tangent to a circle?

Perpendicularity Signup and view all the answers

What is the property of proportion that states wx = yz?

Cross Multiplication Signup and view all the answers

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

Basic Proportionality Theorem Signup and view all the answers

What is the ratio of the sides of a triangle when a line is drawn parallel to one side?

AD/DB = AE/EC Signup and view all the answers

What is the first step in determining the equation of a tangent to a circle?

Write the equation of the circle in standard form Signup and view all the answers

What is the gradient-point form of the equation of a straight line?

y - y1 = m(x - x1) Signup and view all the answers

What is the property of proportion that states x/w = z/y?

Reciprocal Proportion Signup and view all the answers

What is the key concept in solving proportional problems?

All of the above Signup and view all the answers

Which of the following expressions is equivalent to (\cos(\alpha + \beta)) using only the cosine difference formula and the even-odd identities?

(\cos \alpha \cos \beta - \sin \alpha \sin \beta) Signup and view all the answers

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?

(2 - 2 \cos(\alpha - \beta)) Signup and view all the answers

Using the compound angle identities, find the exact value of (\sin 15°) in terms of radicals.

(\frac{\sqrt{6} - \sqrt{2}}{4}) Signup and view all the answers

Which of the following expressions is equivalent to (\sin(\alpha + \beta)) in terms of (\cos(\alpha - \beta)) and the even-odd identities?

(\cos(\alpha - \beta + \frac{\pi}{2})) Signup and view all the answers

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

(\cos(\alpha + \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta) Signup and view all the answers

If (\cos(\alpha - \beta) = \frac{1}{2}) and (\sin(\alpha - \beta) = \frac{\sqrt{3}}{2}), what is the value of (\tan(\alpha - \beta))?

(\sqrt{3}) Signup and view all the answers

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?

(\alpha - ( - \beta)) Signup and view all the answers

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))?

(-\frac{33}{65}) Signup and view all the answers

What relationship holds true if two triangles have their corresponding sides in proportion?

The triangles are similar. Signup and view all the answers

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?

Angle C is equal to angle F. Signup and view all the answers

What can be concluded about the areas of two triangles with equal bases situated between the same parallel lines?

Their areas are equal. Signup and view all the answers

In the proof of the similarity of triangles using the proportionality theorem, what role does the line GH serve?

It is a line drawn parallel to side BC. Signup and view all the answers

What is indicated by the equality of the proportions $rac{AB}{DE} = rac{AC}{DF} = rac{BC}{EF}$?

The triangles are similar. Signup and view all the answers

If the statement of the Converse of the Pythagorean Theorem is true, what can be inferred about the triangle?

It is a right-angled triangle. Signup and view all the answers

Which of the following correctly summarizes the key concept of equiangular triangles?

Their corresponding sides are in proportion. Signup and view all the answers

Which formula must be applied to find the area of a triangle given the base and the height?

Area = rac{1}{2} imes base imes height Signup and view all the answers

In proving that triangles are similar, what geometric criterion is used when establishing that corresponding angles are equal?

Corresponding angles. Signup and view all the answers

How does the Pythagorean theorem relate to right-angled triangles?

It provides a relationship between the sides of right-angled triangles. Signup and view all the answers

What is the simplified result of ( \sin(\alpha - \beta) ) using co-function identities?

( \sin \alpha \cos \beta - \cos \alpha \sin \beta ) Signup and view all the answers

Which formula represents the cosine of a sum?

( \cos(\alpha + \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta ) Signup and view all the answers

Which identity corresponds to the sine of a double angle?

( \sin(2\alpha) = 2 \sin \alpha \cos \alpha ) Signup and view all the answers

What is the equivalent form of the cosine of a double angle if ( \alpha = 30^ ext{circ} )?

( ext{Both (a) and (b)} ) Signup and view all the answers

What is NOT a step in finding general solutions for trigonometric equations?

Verify the solutions using graphical methods. Signup and view all the answers

In the derivation of the cosine of a double angle, what is the initial formula used?

( \cos(\alpha + \alpha) = \cos \alpha \cos \alpha - \sin \alpha \sin \alpha ) Signup and view all the answers

What is the correct expression for ( \cos(2\alpha) ) using only sine?

( 1 - 2 \sin^2 \alpha ) Signup and view all the answers

Which compound angle formula for sine is correct for ( \alpha + \beta )?

( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta ) Signup and view all the answers

Which of the following correctly describes how to use a CAST diagram?

It shows where each trig function is positive or negative based on the angle. Signup and view all the answers

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?

1:2 Signup and view all the answers

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?

F is the midpoint of BC and DE is parallel to BC. Signup and view all the answers

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?

1:4 Signup and view all the answers

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?

3:5 Signup and view all the answers

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?

6 Signup and view all the answers

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?

36 square units Signup and view all the answers

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?

Triangle ABD is congruent to triangle ACD. Signup and view all the answers

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?

10 Signup and view all the answers

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?

32 square units Signup and view all the answers

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?

2:3 Signup and view all the answers

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).

h = \frac{(d - b) \sin \alpha \tan \beta}{\sin (\beta - \alpha)} Signup and view all the answers

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).

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

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.

\frac{35\sqrt{3}}{4} Signup and view all the answers

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.

\sqrt{135^2 + 120^2 - 2 \cdot 135 \cdot 120 \cdot \cos 60^\circ} Signup and view all the answers

A triangular field (ABC) has sides (a = 100) meters, (b = 150) meters, and (c = 200) meters. Calculate the measure of angle (B).

\cos^{-1} (\frac{150^2 + 200^2 - 100^2}{2 \cdot 150 \cdot 200}) Signup and view all the answers

A triangle (ABC) has sides (a = 5), (b = 7), and (c = 8). Calculate the measure of angle (A).

\cos^{-1} (\frac{5^2 + 7^2 - 8^2}{2 \cdot 5 \cdot 7}) Signup and view all the answers

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).

\sqrt{10^2 + 15^2 - 2 \cdot 10 \cdot 15 \cdot \cos 60^\circ} Signup and view all the answers

A triangle (ABC) has sides (a = 6), (b = 8), and (c = 10). Calculate the area of the triangle.

24 Signup and view all the answers

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?

10 Signup and view all the answers

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) )?

-1/2 Signup and view all the answers

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?

19.2 Signup and view all the answers

The equation of a circle is ( x^2 + y^2 - 6x + 8y = 0 ). What is the radius of the circle?

5 Signup and view all the answers

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) )?

x = 7 Signup and view all the answers

The equation of a circle is ( (x + 1)^2 + (y - 4)^2 = 9 ). What are the coordinates of the center of the circle?

(-1, 4) Signup and view all the answers

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 )?

8 Signup and view all the answers

Two ratios, ( a/b ) and ( c/d ), are proportional. Which of the following expressions is NOT necessarily true?

a/b = d/c Signup and view all the answers

A circle has a center at ( (1, -2) ) and passes through the point ( (4, 1) ). What is the equation of the circle?

( (x - 1)^2 + (y + 2)^2 = 18 ) Signup and view all the answers

A tangent line to a circle with center ( (2, 3) ) passes through the point ( (5, 1) ). What is the slope of the tangent line?

-3/2 Signup and view all the answers

Two triangles, ( ABC ) and ( DEF ), are similar. If ( AB = 6 ), ( BC = 8 ), and ( DE = 9 ), what is the length of ( EF )?

12 Signup and view all the answers

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?

7.5 Signup and view all the answers

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?

1:4 Signup and view all the answers

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?

3:5 Signup and view all the answers

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?

1:2 Signup and view all the answers

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?

DE is parallel to BC and DE = 1/2 BC Signup and view all the answers

In the proof of the theorem "Triangles with Sides in Proportion are Similar", which geometric concept is employed to demonstrate that (GH \parallel BC)?

The proportionality theorem Signup and view all the answers

Which of the following correctly summarizes the key concept of equiangular triangles?

Triangles with equal corresponding angles Signup and view all the answers

In the proof of the Pythagorean theorem, which similarity criterion is used to establish the similarity between ( riangle ABD) and ( riangle CBA)?

AAA similarity criterion Signup and view all the answers

In proving that triangles are similar, what geometric criterion is used when establishing that corresponding angles are equal?

Angle-Angle-Angle (AAA) Signup and view all the answers

What is the key concept used to derive the equation (AB^2 = BD \cdot BC) in the proof of the Pythagorean theorem?

Proportionality of corresponding sides in similar triangles Signup and view all the answers

What is the simplified result of ( \sin(\alpha - \beta) ) using co-function identities?

( \cos(\beta - \alpha) ) Signup and view all the answers

In the proof of the Converse of the Pythagorean Theorem, which statement is used to infer that (\angle A) is a right angle?

The Law of Cosines Signup and view all the answers

Which of the following correctly describes how to use a CAST diagram?

To determine the signs of trigonometric ratios in all four quadrants Signup and view all the answers

Which of the following statements accurately describes the relationship between the areas of two triangles with equal bases situated between the same parallel lines?

The areas are equal. Signup and view all the answers

Which formula represents the cosine of a sum?

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

What is the fundamental concept that underlies the relationship between the sides and angles of equiangular triangles?

The proportionality of corresponding sides Signup and view all the answers

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?

All corresponding angles are equal. Signup and view all the answers

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)?

To establish the similarity between ( riangle ABC) and ( riangle DEF) Signup and view all the answers

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?

The triangle is a right-angled triangle Signup and view all the answers

In the proof of the Pythagorean theorem, what is the key role of the line segment (AD) drawn perpendicular to (BC)?

To create similar triangles for applying proportionality Signup and view all the answers

What is the formula for the area of a triangle when no perpendicular height is given?

All of the above Signup and view all the answers

When should the sine rule be used?

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 Signup and view all the answers

What is the formula for the height of a pole?

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

What is the formula for the height of a building?

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

What is the first step in solving problems in three dimensions?

Draw a sketch and consider the given information Signup and view all the answers

What is the purpose of the cosine rule?

To find the length of a side in a triangle when two sides and the included angle are known Signup and view all the answers

What is the formula fortan β in the problem of the height of a pole?

tan β = h / FB Signup and view all the answers

What is the strategy for solving trigonometric problems in three dimensions?

Draw a sketch, identify the relevant triangles, and apply the sine or cosine rules as needed Signup and view all the answers

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

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

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

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

The equation of a circle is given as (x - 3)^2 + (y + 2)^2 = 16. Which of the following points lies on the circle?

(7, -2) Signup and view all the answers

Which of the following is NOT a property of a circle with center at the origin?

Symmetric about the line y = 2x Signup and view all the answers

What is the equation of the circle with center at (-1, 4) and passing through the point (2, 1)?

(x + 1)^2 + (y - 4)^2 = 18 Signup and view all the answers

If the equation of a circle is x^2 + y^2 + 8x - 10y + 25 = 0, what is the value of the radius?

4 Signup and view all the answers

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?

(x - 2)^2 + (y - 2)^2 = 16 Signup and view all the answers

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?

The constant term on the right side of the equation is quadrupled. Signup and view all the answers

What is the compound angle formula for the sine of the sum of two angles?

$ an( heta + eta) = rac{ an heta + an eta}{1 - an heta an eta}$ Signup and view all the answers

Using the co-function identities, what is the correct expression for $ an(90^ ext{circ} - heta)$?

$rac{1}{ an heta}$ Signup and view all the answers

Which of the following represents the cosine of a difference?

$ an( heta - eta) = rac{ an heta + an eta}{1 - an heta an eta}$ Signup and view all the answers

Which alternate form of the cosine double angle formula is derived from the identity $sin^2 heta + cos^2 heta = 1$?

$cos(2 heta) = 2 sin^2 heta - 1$ Signup and view all the answers

What is the correct general solution to the equation $sin(x) = 0$?

$x = n ext{π}$, where n is an integer Signup and view all the answers

From the double angle identity, what is the equivalent expression for $sin(2 heta)$?

$sin(2 heta) = 2 sin heta cos heta$ Signup and view all the answers

Which of these options correctly describes the component identities for sine based on co-function identities?

$sin(90° - heta) = cos heta$ Signup and view all the answers

In solving the equation $cos(x) = rac{1}{2}$, which of the following represents the solution in the general form?

$x = 60^ ext{circ} + n 180^ ext{circ}$ Signup and view all the answers

Which compound angle formula correctly represents $ an( heta + eta)$?

$ an( heta + eta) = rac{ an heta + an eta}{1 + an heta an eta}$ Signup and view all the answers

Using the compound angle identities, find the exact value of (\cos 15^\circ) in terms of radicals.

(\frac{\sqrt{6} + \sqrt{2}}{4}) Signup and view all the answers

Which of the following expressions is equivalent to (\sin(\alpha + \beta)) in terms of (\cos(\alpha - \beta)) and the even-odd identities?

(\cos(\pi/2 - (\alpha - \beta))) Signup and view all the answers

If (\cos(\alpha - \beta) = \frac{1}{2}) and (\sin(\alpha - \beta) = \frac{\sqrt{3}}{2}), what is the value of (\tan(\alpha - \beta)?

(\sqrt{3}) Signup and view all the answers

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?

(\alpha - (-\beta)) Signup and view all the answers

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))?

(\frac{-33}{65}) Signup and view all the answers

Which of the following correctly summarizes the key concept of equiangular triangles?

Equiangular triangles have all angles equal in measure. Signup and view all the answers

In proving that triangles are similar, what geometric criterion is used when establishing that corresponding angles are equal?

Angle-Angle (AA) similarity criterion Signup and view all the answers

What is the simplified result of (\sin(\alpha - \beta)) using co-function identities?

(\cos(\beta - \alpha)) Signup and view all the answers

What is the geometric concept that relates to the equality of ratios of corresponding sides in two triangles?

Proportion Signup and view all the answers

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

They are equal Signup and view all the answers

What is the property of proportion that states wx = yz?

Cross Multiplication Signup and view all the answers

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

y - y1 = m tangent (x - x1) Signup and view all the answers

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

m radius = -1/m tangent Signup and view all the answers

What is the first step in determining the equation of a tangent to a circle?

Write the equation of the circle in standard form Signup and view all the answers

What is the key concept in solving proportional problems?

All of the above Signup and view all the answers

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

Basic Proportionality Theorem Signup and view all the answers

What is the ratio of the sides of a triangle when a line is drawn parallel to one side?

Proportional Signup and view all the answers

What is the application of proportions in geometric figures?

Comparing different parts of the figures Signup and view all the answers

What is the equivalent form of the sine of a difference formula using co-function identities?

sin(α)cos(β) - cos(α)sin(β) Signup and view all the answers

Which of the following is a correct step in finding general solutions for trigonometric equations?

Use the CAST diagram to determine where the function is positive or negative Signup and view all the answers

What is the correct expression for cos(2α) using only sine?

1 - 2sin²(α) Signup and view all the answers

Which compound angle formula for cosine is correct for α - β?

cos(α)cos(β) - sin(α)sin(β) Signup and view all the answers

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

To determine where the function is positive or negative Signup and view all the answers

Which of the following is NOT a step in deriving the cosine of a double angle?

Diving by 2 Signup and view all the answers

What is the correct expression for sin(2α) using cosine?

2sin(α)cos(α) Signup and view all the answers

Which of the following correctly describes the sine of a double angle?

2sin(α)cos(α) Signup and view all the answers

What is the equivalent form of the cosine of a double angle if α = 30°?

1/2 Signup and view all the answers

Which of the following is a correct step in deriving the sine of a double angle?

Letting α = β Signup and view all the answers

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

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

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?

(D/2)^2 Signup and view all the answers

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?

(\cos(\alpha + \beta) - 2\sin\alpha\sin\beta) Signup and view all the answers

What represents the distance from the origin to a point P(x, y) on a circle with radius r?

$OP = r^2$ Signup and view all the answers

If (\cos(\alpha - \beta) = \frac{1}{2}) and (\sin(\alpha - \beta) = \frac{\sqrt{3}}{2}), what is the value of (\tan(\alpha - \beta))?

(\sqrt{3}) Signup and view all the answers

Which statement accurately reflects the symmetry properties of a circle centered at the origin?

It is symmetric about the x-axis, y-axis, and the origin. Signup and view all the answers

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

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

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?

$r^2 = x^2 + y^2$ Signup and view all the answers

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?

(\alpha - (-\beta)) Signup and view all the answers

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?

D and E influence the position of the center of the circle. Signup and view all the answers

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))?

(\frac{-16}{65}) Signup and view all the answers

What relationship holds true if two triangles have their corresponding sides in proportion?

The triangles are similar. Signup and view all the answers

After completing the square, what is the general transformation for the equation of a circle?

It retains its form but shifts center to (a, b). Signup and view all the answers

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

It provides the basis for distance relationships in the circle. Signup and view all the answers

Which of the following correctly summarizes the key concept of equiangular triangles?

Equiangular triangles have corresponding angles equal. Signup and view all the answers

In proving that triangles are similar, what geometric criterion is used when establishing that corresponding angles are equal?

Angle-Angle (AA) Similarity Postulate Signup and view all the answers

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?

7.5 Signup and view all the answers

In the proof of the proportionality theorem, why is line segment GH constructed parallel to BC?

To create similar triangles AGH and ABC, thus proving the proportionality of sides. Signup and view all the answers

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)?

The similarity establishes that the corresponding sides are proportional, leading to the equation (AB/BC = BD/AB). Signup and view all the answers

What is the primary implication of the statement (rac{AB}{DE} = rac{AC}{DF} = rac{BC}{EF}) in relation to triangles ABC and DEF?

It indicates that the triangles are similar. Signup and view all the answers

Which of the following statements is NOT a valid condition for determining similarity of triangles?

Two triangles are similar if they have the same area. Signup and view all the answers

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?

The triangles have equal heights. Signup and view all the answers

What is the main difference between the Pythagorean Theorem and its converse?

The Pythagorean Theorem proves that the square of the hypotenuse is equal to the sum of squares of other sides, while the converse proves that if the sum of squares of two sides is equal to the square of the third side, then the triangle is right-angled. Signup and view all the answers

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)?

To prove that the triangles ABD and CBA are similar. Signup and view all the answers

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

The area of a triangle is equal to half the product of its base and height. Signup and view all the answers

What is the key concept that distinguishes the similarity of triangles from their congruence?

Similar triangles have the same shape but different sizes, while congruent triangles have the same shape and size. Signup and view all the answers

In the proof of the Pythagorean Theorem, why is it necessary to construct AD perpendicular to BC?

To create two smaller right-angled triangles, ABD and CAD, which are similar to triangle ABC. Signup and view all the answers

What is the relationship between the heights of two triangles on the same base and equal in area?

The heights are equal. Signup and view all the answers

Which of the following conditions must be satisfied for two polygons to be considered similar?

Only pairs of corresponding lengths must be proportional. Signup and view all the answers

What does the Mid-point Theorem state regarding the sides of a triangle?

The line joining the midpoints is parallel and half as long as the third side. Signup and view all the answers

What conclusion can be drawn from the statement 'If two triangles are equiangular, then their corresponding sides are in proportion'?

The triangles are similar. Signup and view all the answers

What must be true for two triangles to be proven similar by the Proportion Theorem?

The ratios of the lengths of the sides must be equal. Signup and view all the answers

Which of the following statements about the areas of triangles situated between the same parallel lines is true?

Their areas are proportionate to the lengths of their bases. Signup and view all the answers

In the context of polygons, what does the term 'congruent' imply?

The polygons are of equal shape and size. Signup and view all the answers

What logical step is involved in proving two triangles similar using equiangular criteria?

Showing that corresponding sides are in proportion. Signup and view all the answers

Which of the following formulas represents the relationship of areas for triangles having equal bases between two parallel lines?

Area = 1/2 × base × height Signup and view all the answers

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?

Area Rule Signup and view all the answers

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?

The corresponding sides are proportional Signup and view all the answers

What is the purpose of using the Sine Rule in trigonometry?

To find the length of a side of a triangle, given two sides and an angle (not the included angle) Signup and view all the answers

In a triangle ABC, if angle A is 30 degrees and side a = 5, what is the value of side b?

5√3 Signup and view all the answers

What is the formula for the area of a triangle, given two sides and the included angle?

1/2 ab sin C Signup and view all the answers

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?

The triangles are similar Signup and view all the answers

What is the purpose of using the Cosine Rule in trigonometry?

To find the length of a side of a triangle, given two sides and the included angle Signup and view all the answers

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?

d tan α Signup and view all the answers

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?

(a, b) Signup and view all the answers

What is the relationship between the gradient of the radius and the gradient of the tangent in a circle?

m_radius * m_tangent = -1 Signup and view all the answers

If two ratios are equal, what is the relationship between the quantities involved?

Proportional Signup and view all the answers

What is the key concept in determining the equation of a tangent to a circle?

Perpendicularity Signup and view all the answers

What is the equation of a tangent to a circle in the gradient-point form?

y - y1 = m_tangent (x - x1) Signup and view all the answers

What is the property of proportion that states wx = yz?

Cross Multiplication Signup and view all the answers

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

Basic Proportionality Theorem Signup and view all the answers

What is the ratio of the sides of a triangle when a line is drawn parallel to one side?

Proportional Signup and view all the answers

What is the first step in determining the equation of a tangent to a circle?

Find the center of the circle Signup and view all the answers

What is the purpose of using proportions in geometry?

To compare different parts of geometric figures Signup and view all the answers

If two triangles have their corresponding sides in proportion, which of the following statements is true?

They are similar Signup and view all the answers

What is the purpose of Thales' theorem in geometry?

To divide a line into proportional parts Signup and view all the answers

If two triangles are equiangular, which of the following is true?

Their corresponding sides are proportional Signup and view all the answers

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?

Basic Proportionality theorem Signup and view all the answers

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?

They are proportional Signup and view all the answers

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?

Pythagorean theorem Signup and view all the answers

If two triangles have equal bases and are situated between the same parallel lines, what can be concluded about their areas?

They are equal Signup and view all the answers

What is the formula for the area of a parallelogram?

Base x Height Signup and view all the answers

If two triangles are similar, which of the following is true?

Their corresponding sides are proportional Signup and view all the answers

What is the purpose of the Triangle Proportionality theorem?

To establish similarity between triangles Signup and view all the answers

Which of the following formulas is used to derive the sine of a double angle?

sin(2α) = 2 sin α cos α Signup and view all the answers

Given the equation of a circle (x^2 + y^2 - 6x + 4y - 12 = 0), what is the radius of the circle?

5 Signup and view all the answers

What is the equivalent form of the cosine of a double angle if α = 45°?

cos(2α) = 2 cos^2 α - 1 Signup and view all the answers

What is the equation of the circle with center at ((-2, 3)) and passing through the point ((1, -1))?

( (x + 2)^2 + (y - 3)^2 = 20 ) Signup and view all the answers

Which of the following is NOT a step in finding general solutions for trigonometric equations?

Plot the graph of the function Signup and view all the answers

What is the initial formula used in the derivation of the cosine of a double angle?

cos(α + β) = cos α cos β - sin α sin β Signup and view all the answers

If the equation of a circle is (x^2 + y^2 + 10x - 8y + 25 = 0), what is the center of the circle?

(-5, 4) Signup and view all the answers

Which identity corresponds to the cosine of a double angle?

cos(2α) = cos^2 α - sin^2 α Signup and view all the answers

Which of the following points lies on the circle with the equation (x^2 + y^2 - 8x + 6y = 0)?

(4, -3) Signup and view all the answers

What is the correct expression for cos(2α) using only sine?

cos(2α) = 1 - 2 sin^2 α Signup and view all the answers

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))?

(y = - rac{4}{3}x + rac{29}{3}) Signup and view all the answers

Which compound angle formula for sine is correct for α + β?

sin(α + β) = sin α cos β + cos α sin β Signup and view all the answers

The equation of a circle is (x^2 + y^2 - 10x + 8y + 16 = 0). What is the length of the diameter of this circle?

12 Signup and view all the answers

Which of the following correctly describes how to use a CAST diagram?

Determine where the function is positive or negative using the CAST diagram Signup and view all the answers

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?

(y = 2x - 7) Signup and view all the answers

In a triangle, if the sine of an angle is x, what is the general solution for the angle?

θ = sin^-1 x + k ⋅ 180° Signup and view all the answers

What is the formula for the area of a triangle, given two sides and the included angle?

Area = (1/2)bc sin A Signup and view all the answers

What is the simplified result of sin(α - β) using co-function identities?

sin(α - β) = sin α cos β - cos α sin β Signup and view all the answers

What is the equation of the circle with diameter endpoints at ((2, 5)) and ((8, 1))?

( (x - 5)^2 + (y - 3)^2 = 13 ) Signup and view all the answers

When should the cosine rule be used?

When no right angle is given, and either two sides and an angle or three sides are given. Signup and view all the answers

Which formula represents the cosine of a sum?

cos(α + β) = cos α cos β - sin α sin β Signup and view all the answers

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?

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

What is the first step in solving problems in three dimensions?

Draw a sketch of the problem. Signup and view all the answers

What is the formula for the area of a triangle, given the base and height?

Area = (1/2)bh Signup and view all the answers

When should the sine rule be used?

When no right angle is given, and either two sides and the included angle or two angles and a side are given. Signup and view all the answers

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?

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

In the proof of the Pythagorean Theorem, why is it essential to draw a perpendicular line (AD) from (A) to (BC)?

To create similar triangles (ABD) and (CAD) to establish a relationship between the sides. Signup and view all the answers

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)?

To create two similar triangles (AGH) and (ABC) and establish the equality of corresponding angles. Signup and view all the answers

What is the significance of the statement 'Triangles with equal heights have areas proportional to their bases' in the context of triangles and proportionality?

It implies that the ratio of the areas of two triangles with equal heights is equal to the ratio of their bases. Signup and view all the answers

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.'?

Two triangles with proportional sides are always congruent. Signup and view all the answers

In the context of the Pythagorean Theorem, what is the key relationship between the similar triangles (ABD), (CAD), and (CBA)?

Their corresponding sides are in proportion. Signup and view all the answers

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?

The height of ( riangle ABC) is twice the height of ( riangle DEF). Signup and view all the answers

What is the most accurate interpretation of the converse of the Pythagorean Theorem?

If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. Signup and view all the answers

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)?

7.5 Signup and view all the answers

What is the primary function of the 'SSS' similarity criterion in the proof of the Pythagorean Theorem?

To demonstrate that triangles (ABD) and (CAD) are congruent. Signup and view all the answers

Which of the following concepts is NOT directly related to the concept of 'proportionality in triangles'?

The Pythagorean Theorem. Signup and view all the answers

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)?

16 Signup and view all the answers

If two triangles have corresponding sides in proportion, what can you definitively conclude about the triangles?

They are similar. Signup and view all the answers

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)?

6.67 Signup and view all the answers

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)?

The area of (ABC) is equal to the sum of the areas of (ABD) and (CAD). Signup and view all the answers

If two triangles are similar, which of the following statements is ALWAYS TRUE?

The corresponding angles of the triangles are equal. Signup and view all the answers

Given that ( riangle ABC \sim riangle DEF), with (AB = 5), (BC = 7), and (DF = 10), what is the length of (EF)?

14 Signup and view all the answers

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)?

DE is half the length of BC. Signup and view all the answers

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)?

6 Signup and view all the answers

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)?

3:1 Signup and view all the answers

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)?

1:4 Signup and view all the answers

Given that (\cos(\alpha - \beta) = \frac{1}{2}) and (\sin(\alpha - \beta) = \frac{\sqrt{3}}{2}), what is the value of (\tan(\alpha - \beta))?

$\sqrt{3}$ Signup and view all the answers

Which of the following expressions is equivalent to (\sin(\alpha + \beta)) in terms of (\cos(\alpha - \beta)) and the even-odd identities?

$-\cos(\alpha - \beta)$ Signup and view all the answers

Which of the following expressions is equivalent to (\cos(\alpha + \beta)) using only the cosine difference formula and the even-odd identities?

$\cos \alpha \cos \beta - \sin \alpha \sin \beta$ Signup and view all the answers

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?

$\cos^2 \alpha - 2 \cos \alpha \cos \beta + \cos^2 \beta + \sin^2 \alpha - 2 \sin \alpha \sin \beta + \sin^2 \beta$ Signup and view all the answers

Using the compound angle identities, find the exact value of (\sin 15°) in terms of radicals.

$\frac{\sqrt{6} + \sqrt{2}}{4}$ Signup and view all the answers

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?

$\alpha - (-\beta)$ Signup and view all the answers

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))?

$-\frac{16}{65}$ Signup and view all the answers

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

$\cos(\alpha + \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$ Signup and view all the answers

Gr12 Mathematics: June Hard P(2)

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)?

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

What is the purpose of completing the square in the context of circle equations?

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

What is the first step in completing the square to find the center and radius of a circle?

What is the equation derived from the Pythagorean theorem?

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

What is the purpose of using the distance formula in the context of circle equations?

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

What is the key concept in determining the equation of a tangent to a circle?

What is the property of proportion that states wx = yz?

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 ratio of the sides of a triangle when a line is drawn parallel to one side?

What is the first step in determining the equation of a tangent to a circle?

What is the gradient-point form of the equation of a straight line?

What is the property of proportion that states x/w = z/y?

What is the key concept in solving proportional problems?

Which of the following expressions is equivalent to (\cos(\alpha + \beta)) using only the cosine difference formula and the even-odd identities?

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?

Using the compound angle identities, find the exact value of (\sin 15°) in terms of radicals.

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 is NOT a valid compound angle identity?

If (\cos(\alpha - \beta) = \frac{1}{2}) and (\sin(\alpha - \beta) = \frac{\sqrt{3}}{2}), what is the value of (\tan(\alpha - \beta))?

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?

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))?

What relationship holds true if two triangles have their corresponding sides in proportion?

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?

What can be concluded about the areas of two triangles with equal bases situated between the same parallel lines?

In the proof of the similarity of triangles using the proportionality theorem, what role does the line GH serve?

What is indicated by the equality of the proportions $rac{AB}{DE} = rac{AC}{DF} = rac{BC}{EF}$?

If the statement of the Converse of the Pythagorean Theorem is true, what can be inferred about the triangle?

Which of the following correctly summarizes the key concept of equiangular triangles?

Which formula must be applied to find the area of a triangle given the base and the height?

In proving that triangles are similar, what geometric criterion is used when establishing that corresponding angles are equal?

How does the Pythagorean theorem relate to right-angled triangles?

What is the simplified result of ( \sin(\alpha - \beta) ) using co-function identities?

Which formula represents the cosine of a sum?

Which identity corresponds to the sine of a double angle?

What is the equivalent form of the cosine of a double angle if ( \alpha = 30^ ext{circ} )?

What is NOT a step in finding general solutions for trigonometric equations?

In the derivation of the cosine of a double angle, what is the initial formula used?

What is the correct expression for ( \cos(2\alpha) ) using only sine?

Which compound angle formula for sine is correct for ( \alpha + \beta )?

Which of the following correctly describes how to use a CAST diagram?

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?

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?

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?

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 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?

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?

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, points D and E are the midpoints of sides AB and AC respectively. If DE = 5, what is the length of BC?

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?

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?

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.

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.

A triangular field (ABC) has sides (a = 100) meters, (b = 150) meters, and (c = 200) meters. Calculate the measure of angle (B).

A triangle (ABC) has sides (a = 5), (b = 7), and (c = 8). Calculate the measure of angle (A).

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 triangle (ABC) has sides (a = 6), (b = 8), and (c = 10). Calculate the area of the triangle.

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?

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) )?

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?

The equation of a circle is ( x^2 + y^2 - 6x + 8y = 0 ). What is the radius of the circle?

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) )?

The equation of a circle is ( (x + 1)^2 + (y - 4)^2 = 9 ). What are the coordinates of the center of the circle?

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 )?

Two ratios, ( a/b ) and ( c/d ), are proportional. Which of the following expressions is NOT necessarily true?

A circle has a center at ( (1, -2) ) and passes through the point ( (4, 1) ). What is the equation of the circle?

A tangent line to a circle with center ( (2, 3) ) passes through the point ( (5, 1) ). What is the slope of the tangent line?

Two triangles, ( ABC ) and ( DEF ), are similar. If ( AB = 6 ), ( BC = 8 ), and ( DE = 9 ), what is the length of ( EF )?

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?

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?

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?

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?

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?