Geometry and Trigonometry Practice Questions — SAT — Page 4

Question 61Question →

In triangle $ABC$ , point $D$ lies on side $BC$ such that the ratio of $BD$ to $DC$ is $1$ to $2$ . Point $E$ lies on segment $AD$ such that the ratio of $AE$ to $ED$ is $3$ to $1$ . A line passing through $E$ and parallel to $AC$ intersects side $AB$ at $F$ and side $BC$ at $G$ . If the area of triangle $ABC$ is $128$ , what is the area of triangle $BFG$ ?

Show answer & explanation

Answer: 32

Answer

32

The correct answer is 32. By establishing a coordinate system, we find that the similarity ratio of triangle

BFG

to triangle

ABC

is exactly

1/2

regardless of the triangle's shape. Since the ratio of the areas of similar triangles is the square of the similarity ratio, the area of triangle

BFG

is

(1/2)^2 = 1/4

of the area of triangle

ABC

, which is

128 \times 1/4 = 32

.

Step-by-Step Solution

1

Set up a coordinate system to represent the triangle's vertices.

Let

B = (0, 0)

and

C = (3, 0)

. Since

D

lies on side

BC

and

BD:DC = 1:2

, the coordinates of

D

are

(1, 0)

. Let

A = (a, b)

.

Setting up coordinates simplifies the proof by allowing algebraic verification of the ratio.

2

Calculate the coordinates of point

E

on segment

AD

.

Using the section formula with ratio

AE:ED = 3:1

,

E = \left(\frac{1(a) + 3(1)}{4}, \frac{1(b) + 3(0)}{4}\right) = \left(\frac{a+3}{4}, \frac{b}{4}\right)

.

Finding the coordinates of

E

is necessary to determine the equation of line

FG

.

3

Find the equation of line

FG

which is parallel to

AC

and passes through

E

.

The slope of

AC

is

m = \frac{-b}{3-a}

. The equation of

FG

is

y - \frac{b}{4} = \frac{-b}{3-a}\left(x - \frac{a+3}{4}\right)

.

The line

FG

is parallel to

AC

, meaning they share the same slope.

4

Find the coordinates of

G

by setting

y = 0

in the equation of

FG

.

Setting

y = 0

yields

-\frac{b}{4} = \frac{-b}{3-a}\left(x_G - \frac{a+3}{4}\right) \implies x_G = 1.5

. Thus,

G = (1.5, 0)

.

Point

G

is the intersection of the line

FG

with side

BC

(the x-axis).

5

Determine the similarity ratio and calculate the area of triangle

BFG

.

Since

G

is the midpoint of

BC

, the similarity ratio of

\triangle BFG

to

\triangle BAC

is

k = \frac{1}{2}

. The ratio of their areas is

k^2 = \frac{1}{4}

. The area of

\triangle BFG

is

128 \times \frac{1}{4} = 32

.

Similar triangles have area ratios equal to the square of their similarity ratio.

Key Concept

The ratio of the areas of similar triangles is equal to the square of their similarity ratio.

Alternative Method

Using Menelaus's Theorem on triangle

ABD

and transversal

FEG

can also establish the midpoint relations directly without using coordinates.

Estimated Time:3m 0s

Question 62Question →

An aerospace company designs two similar solar panels. The smaller panel is in the shape of a parallelogram and has an area of $80$ square centimeters. If the perimeter of the larger panel is $1.5$ times the perimeter of the smaller panel, what is the area, in square centimeters, of the larger panel?

120

200

180

270

Show answer & explanation

Answer: 180

Answer

180

Since the two solar panels are similar, the ratio of their perimeters is equal to their linear scale factor,

k = 1.5

. The ratio of their areas is equal to the square of the linear scale factor,

k^2 = (1.5)^2 = 2.25

. Therefore, the area of the larger panel is the product of the smaller panel's area and the area scale factor:

80 \times 2.25 = 180

square centimeters.

Step-by-Step Solution

1

Determine the linear scale factor between the two similar solar panels.

The linear scale factor is

k = 1.5

.

For similar figures, the ratio of any corresponding linear measurements (such as perimeters) is equal to the linear scale factor. Here, the perimeter of the larger panel is

1.5

times that of the smaller panel.

2

Calculate the area scale factor by squaring the linear scale factor.

The area scale factor is

k^2 = (1.5)^2 = 2.25

.

The ratio of the areas of two similar two-dimensional shapes is equal to the square of their linear scale factor.

3

Multiply the area of the smaller solar panel by the area scale factor to find the area of the larger panel.

The area of the larger solar panel is

80 \times 2.25 = 180

square centimeters.

Multiplying the original area by the area scale factor yields the scaled area of the larger similar figure.

Key Concept

Area scaling of similar figures

Question 63Question →

An acute angle $\theta$ satisfies the equation $\sin(\theta) = 0.6$ . What is the value of $\cos(\theta)$ ?

Show answer & explanation

Answer: 0.8

Answer

The value of

\cos(\theta)

is 0.8.

Using the fundamental Pythagorean trigonometric identity

\sin^2(\theta) + \cos^2(\theta) = 1

, we substitute the given value

\sin(\theta) = 0.6

to obtain

(0.6)^2 + \cos^2(\theta) = 1

, which simplifies to

0.36 + \cos^2(\theta) = 1

. Subtracting

0.36

from both sides yields

\cos^2(\theta) = 0.64

. Taking the positive square root because

\theta

is an acute angle gives

\cos(\theta) = 0.8

.

Step-by-Step Solution

1

State the Pythagorean trigonometric identity

\sin^2(\theta) + \cos^2(\theta) = 1

This identity relates the sine and cosine of any angle.

2

Substitute the value of

\sin(\theta)

into the identity

(0.6)^2 + \cos^2(\theta) = 1

We are given that

\sin(\theta) = 0.6

.

3

Solve for

\cos^2(\theta)

\cos^2(\theta) = 0.64

Subtracting

0.36

from

1

isolates the squared cosine term.

4

Take the square root of both sides

\cos(\theta) = 0.8

Since

\theta

is an acute angle, the value of

\cos(\theta)

must be positive.

Key Concept

Pythagorean Identity

Question 64Question →

A right triangle in the $xy$ -plane has vertices at the coordinates $(0, 0)$ , $(15, 0)$ , and $(15, 8)$ . What is the perimeter of this triangle?

17

23

40

60

Show answer & explanation

Answer: 40

Answer

The perimeter of the triangle is 40.

The perimeter of the triangle is the sum of its three side lengths. The horizontal leg has a length of

15

(from

(0,0)

to

(15,0)

), and the vertical leg has a length of

8

(from

(15,0)

to

(15,8)

). Using the Pythagorean theorem, the hypotenuse has a length of

\sqrt{15^2 + 8^2} = \sqrt{225 + 64} = 17

. Summing these three lengths gives a perimeter of

15 + 8 + 17 = 40

.

Step-by-Step Solution

1

Identify the lengths of the two perpendicular legs of the right triangle from the given coordinates.

The horizontal leg lies on the x-axis from

(0, 0)

to

(15, 0)

, which has a length of

15 - 0 = 15

. The vertical leg runs parallel to the y-axis from

(15, 0)

to

(15, 8)

, which has a length of

8 - 0 = 8

.

The length of horizontal and vertical segments in a coordinate plane can be found by taking the absolute difference of their non-matching coordinates.

2

Use the Pythagorean theorem to calculate the length of the hypotenuse.

The hypotenuse length is

\sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17

.

The Pythagorean theorem states that

a^2 + b^2 = c^2

for any right triangle with legs

a

and

b

and hypotenuse

c

.

3

Calculate the perimeter of the triangle by summing the lengths of all three sides.

15 + 8 + 17 = 40

.

The perimeter of a triangle is the total distance around the boundary, which equals the sum of its three side lengths.

Key Concept

Calculating the perimeter of a right triangle in the coordinate plane by using the Pythagorean theorem to find the hypotenuse.

Question 65Question →

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ . If $\cos(A) = 3\sin(A)$ , what is the value of $\tan(B)$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is

3

.

Because angle

C

is

90^\circ

in right triangle

ABC

, the acute angles

A

and

B

are complementary (

A + B = 90^\circ

). By the co-function identities,

\sin(B) = \cos(A)

and

\cos(B) = \sin(A)

. The tangent of

B

is defined as

\tan(B) = \frac{\sin(B)}{\cos(B)}

. Substituting the co-function identities gives

\tan(B) = \frac{\cos(A)}{\sin(A)}

. Since we are given that

\cos(A) = 3\sin(A)

, we substitute this expression into the numerator to get

\tan(B) = \frac{3\sin(A)}{\sin(A)} = 3

.

Step-by-Step Solution

1

Determine the relationship between the acute angles in right triangle

ABC

.

\sin(B) = \cos(A)

and

\cos(B) = \sin(A)

Since angle

C

is

90^\circ

, the other two angles

A

and

B

must sum to

90^\circ

(they are complementary angles).

2

Express

\tan(B)

in terms of the trigonometric ratios of angle

A

.

\tan(B) = \frac{\cos(A)}{\sin(A)}

By definition, the tangent of angle

B

is the ratio of its sine to its cosine, which yields

\frac{\cos(A)}{\sin(A)}

after substituting the complementary angle relations.

3

Substitute the given relation

\cos(A) = 3\sin(A)

into the expression for

\tan(B)

.

\tan(B) = \frac{3\sin(A)}{\sin(A)}

Substituting the value of

\cos(A)

allows us to simplify the fraction by expressing both terms with

\sin(A)

.

4

Simplify the fraction to get the final numerical value.

\tan(B) = 3

The term

\sin(A)

cancels out from the numerator and denominator since

A

is an acute angle and

\sin(A) \neq 0

.

Key Concept

Co-function identities and trigonometric definitions in a right triangle.

Alternative Method

Alternatively, construct a right triangle where the side opposite to angle

A

has length

1

. Since

\cos(A) = 3\sin(A)

, the ratio of the adjacent side to the hypotenuse is

3

times the ratio of the opposite side to the hypotenuse, meaning the side adjacent to angle

A

must have length

3

. Because angle

B

is the complement of angle

A

, the side opposite to angle

B

is the side adjacent to angle

A

(which is

3

), and the side adjacent to angle

B

is the side opposite to angle

A

(which is

1

). Therefore,

\tan(B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{1} = 3

.

Estimated Time:1m 30s

Question 66Question →

In rectangle $ABCD$ , the length of side $AB$ is $12$ and the length of side $BC$ is $9$ . Point $P$ lies on the diagonal $AC$ such that $AP = \frac{1}{3} AC$ . What is the area of triangle $BPD$ ?

Show answer & explanation

Answer: 18

Answer

18

The correct answer is 18. The area of the right triangle

ABD

is half of the area of rectangle

ABCD

, which is

\frac{12 \times 9}{2} = 54

. Since point

P

lies on diagonal

AC

such that

AP = \frac{1}{3} AC

, triangle

ABP

has a base of

AP

along line

AC

and shares vertex

B

with triangle

ABC

. Thus, its area is

\frac{1}{3}

of the area of triangle

ABC

, which is

\frac{54}{3} = 18

. Similarly, triangle

ADP

shares vertex

D

with triangle

ADC

and has base

AP

, so its area is

\frac{1}{3}

of the area of triangle

ADC

, which is

\frac{54}{3} = 18

. Because

AP = \frac{1}{3} AC < \frac{1}{2} AC

, point

P

lies inside triangle

ABD

. Therefore, the area of triangle

BPD

is the area of triangle

ABD

minus the areas of triangles

ABP

and

ADP

, which is

54 - 18 - 18 = 18

.

Step-by-Step Solution

1

Calculate the area of triangle

ABD

.

\text{Area}(\triangle ABD) = \frac{12 \times 9}{2} = 54

The diagonal

BD

divides the rectangle

ABCD

into two congruent right triangles, each with an area equal to half of the rectangle's total area.

2

Find the areas of triangles

ABP

and

ADP

using the ratio of

AP

to

AC

.

\text{Area}(\triangle ABP) = \frac{1}{3} \times \text{Area}(\triangle ABC) = 18

and

\text{Area}(\triangle ADP) = \frac{1}{3} \times \text{Area}(\triangle ADC) = 18

.

Triangles

ABP

and

ABC

share the same altitude from vertex

B

to the line containing diagonal

AC

. Therefore, the ratio of their areas is equal to the ratio of their bases, which is

\frac{AP}{AC} = \frac{1}{3}

. The same logic applies to triangles

ADP

and

ADC

with vertex

D

.

3

Subtract the areas of triangles

ABP

and

ADP

from the area of triangle

ABD

to find the area of triangle

BPD

.

\text{Area}(\triangle BPD) = 54 - 18 - 18 = 18

Since

AP = \frac{1}{3} AC

, which is less than half the length of the diagonal, point

P

lies strictly within the interior of triangle

ABD

. Thus, the area of triangle

ABD

is partitioned into the areas of triangles

ABP

,

ADP

, and

BPD

.

Key Concept

Partitioning the area of a polygon and using the ratio of bases for triangles sharing a vertex to compute sub-areas.

Alternative Method

Alternatively, place the rectangle in a coordinate system with

B

at the origin

(0,0)

,

C

at

(12,0)

,

A

at

(0,9)

, and

D

at

(12,9)

. The coordinates of point

P

on diagonal

AC

(from

(0,9)

to

(12,0)

) at one-third of the distance from

A

to

C

are

x = 0 + \frac{1}{3}(12 - 0) = 4

and

y = 9 + \frac{1}{3}(0 - 9) = 6

. The area of triangle

BPD

with vertices

B(0,0)

,

P(4,6)

, and

D(12,9)

can be found using the shoelace formula:

\text{Area} = \frac{1}{2} |0(6 - 9) + 4(9 - 0) + 12(0 - 6)| = \frac{1}{2} |36 - 72| = 18

.

Estimated Time:2m 0s

Question 67Question →

In right triangle $RST$ , the measure of angle $T$ is $90^\circ$ and the length of the hypotenuse $RS$ is $10$ . If $\tan(R) + \tan(S) = \frac{5}{2}$ , what is the area of triangle $RST$ ?

10

20

40

50

Show answer & explanation

Answer: 20

Answer

20

Let the lengths of the legs of right triangle

RST

be

RT = x

and

ST = y

. The tangent ratios are

\tan(R) = \frac{y}{x}

and

\tan(S) = \frac{x}{y}

. Substituting these into the given equation gives

\frac{y}{x} + \frac{x}{y} = \frac{5}{2}

, which simplifies to

\frac{x^2 + y^2}{xy} = \frac{5}{2}

. By the Pythagorean theorem,

x^2 + y^2 = 10^2 = 100

. Thus,

\frac{100}{xy} = \frac{5}{2}

, which solves to

xy = 40

. The area of the triangle is

\frac{1}{2}xy = \frac{1}{2}(40) = 20

.

Step-by-Step Solution

1

Express the tangent of angles

R

and

S

in terms of the leg lengths

RT

and

ST

.

\tan(R) = \frac{ST}{RT}

and

\tan(S) = \frac{RT}{ST}

.

By definition, the tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent leg.

2

Substitute these expressions into the given equation and simplify using the Pythagorean theorem.

\frac{ST}{RT} + \frac{RT}{ST} = \frac{ST^2 + RT^2}{RT \cdot ST} = \frac{100}{RT \cdot ST} = \frac{5}{2}

.

Finding a common denominator yields the sum of the squares of the legs in the numerator, which equals the square of the hypotenuse (

RS^2 = 10^2 = 100

).

3

Solve for the product of the legs

RT \cdot ST

and calculate the area of the triangle.

RT \cdot ST = 40

, so

\text{Area} = \frac{1}{2}(RT \cdot ST) = \frac{1}{2}(40) = 20

.

The area of a right triangle is half the product of its perpendicular legs.

Key Concept

Using trigonometric ratios, complementary angles, and the Pythagorean theorem to calculate the area of a right triangle.

Question 68Question →

In right triangle $ABC$ , the hypotenuse $AC$ has length $25$ , and leg $AB$ has length $20$ . What is the length of leg $BC$ ?

Show answer & explanation

Answer: 15

Answer

15

For any right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse (

a^2 + b^2 = c^2

). In right triangle

ABC

, the hypotenuse is

AC = 25

and one of the legs is

AB = 20

. Substituting these values into the theorem gives

20^2 + BC^2 = 25^2

, which simplifies to

400 + BC^2 = 625

. Subtracting

400

from both sides results in

BC^2 = 225

. Taking the square root of both sides gives the length of leg

BC

as

15

.

Step-by-Step Solution

1

Set up the Pythagorean theorem equation for the right triangle.

AB^2 + BC^2 = AC^2

The Pythagorean theorem relates the lengths of the legs and the hypotenuse of any right triangle.

2

Substitute the known lengths of side

AB

and hypotenuse

AC

.

20^2 + BC^2 = 25^2

We plug in

20

for leg

AB

and

25

for hypotenuse

AC

.

3

Evaluate the squared terms.

400 + BC^2 = 625

Squaring

20

gives

400

, and squaring

25

gives

625

.

4

Isolate the variable term

BC^2

.

BC^2 = 225

Subtracting

400

from both sides of the equation yields

225

.

5

Find the length of side

BC

.

BC = 15

Taking the square root of

225

gives the positive length of leg

BC

.

Key Concept

Applying the Pythagorean theorem to calculate a missing leg length of a right triangle when the hypotenuse and one leg length are given.

Question 69Question →

An angle has a measure of $72^\circ$ . What is the measure of this angle in radians?

\frac{2\pi}{5}

\frac{3\pi}{5}

\frac{5\pi}{2}

\frac{\pi}{5}

Show answer & explanation

Answer: $\frac{2\pi}{5}$

Answer

The correct answer is

\frac{2\pi}{5}

radians.

To convert an angle from degrees to radians, multiply the degree measure by

\frac{\pi}{180}

. Doing so for

72^\circ

yields

\frac{72\pi}{180}

. Dividing the numerator and denominator by their greatest common divisor, 36, simplifies the fraction to

\frac{2\pi}{5}

.

Step-by-Step Solution

1

Set up the conversion from degrees to radians by multiplying the given degree measure by

\frac{\pi}{180}

.

\theta = 72 \times \frac{\pi}{180} = \frac{72\pi}{180}

The conversion factor from degrees to radians is

\frac{\pi \text{ radians}}{180^\circ}

.

2

Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, 36.

\frac{72 \div 36}{180 \div 36}\pi = \frac{2\pi}{5}

Simplifying the fraction expresses the angle in standard reduced radian form.

Key Concept

Converting degrees to radians

Question 70Question →

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ , $AC = 15$ , and $BC = 20$ . Altitude $CD$ is drawn to the hypotenuse $AB$ . From point $D$ , perpendicular line segments $DE$ and $DF$ are drawn to sides $AC$ and $BC$ , respectively, where $E$ lies on $AC$ and $F$ lies on $BC$ . What is the ratio of the area of triangle $CEF$ to the area of triangle $ABC$ ?

\frac{144}{625}

\frac{12}{25}

\frac{9}{25}

\frac{16}{25}

Show answer & explanation

Answer: $\frac{144}{625}$

Answer

The ratio of the area of triangle

CEF

to the area of triangle

ABC

is

\frac{144}{625}

.

The correct answer is the fraction representing the ratio of the area of the smaller triangle to the larger triangle. This ratio can be computed by calculating the lengths of the legs of the right triangle

CEF

using similarity relationships, yielding

CE = 9.6

and

CF = 7.2

, which gives an area of

34.56

. Dividing this by the area of triangle

ABC

(

150

) yields the fraction

\frac{144}{625}

. Alternatively, the ratio is equal to the square of the ratio of the altitude to the hypotenuse,

(\frac{CD}{AB})^2 = (\frac{12}{25})^2 = \frac{144}{625}

.

Step-by-Step Solution

1

Use the Pythagorean theorem to calculate the hypotenuse

AB

of the right triangle

ABC

.

AB = \sqrt{AC^2 + BC^2} = \sqrt{15^2 + 20^2} = \sqrt{225 + 400} = \sqrt{625} = 25

Knowing the hypotenuse allows us to calculate the altitude and set up similarity ratios.

2

Calculate the area of triangle

ABC

.

\text{Area}(ABC) = \frac{1}{2} \times AC \times BC = \frac{1}{2} \times 15 \times 20 = 150

This establishes the base area to compare with the area of the smaller triangle.

3

Calculate the length of altitude

CD

by relating the area using base

AB

.

150 = \frac{1}{2} \times AB \times CD \Rightarrow 150 = \frac{1}{2} \times 25 \times CD \Rightarrow CD = 12

The altitude is a key dimension that relates the nested geometry to the main triangle.

4

Identify the properties of the quadrilateral

DECF

and the triangle

CEF

.

DECF

is a rectangle with diagonals

CD = EF = 12

, and

\triangle CEF

is a right triangle with legs

CE

and

CF

.

Since three angles are

90^\circ

(at

C

,

E

, and

F

), the figure is a rectangle, meaning its diagonals are congruent.

5

Apply similarity properties to find the lengths of

CE

and

CF

.

CE = 9.6

and

CF = 7.2

.

Using similar right triangles,

CE = AC - AE = 15 - 5.4 = 9.6

and

CF = BC - BF = 20 - 12.8 = 7.2

.

6

Calculate the area of right triangle

CEF

and find the ratio.

\text{Area}(CEF) = \frac{1}{2} \times 9.6 \times 7.2 = 34.56

. The ratio is

\frac{34.56}{150} = \frac{144}{625}

. Alternatively, the ratio is

\left(\frac{CD}{AB}\right)^2 = \left(\frac{12}{25}\right)^2 = \frac{144}{625}

.

Comparing the two areas gives the final required ratio.

Key Concept

Area ratios and side lengths in similar right triangles using perpendicular projections.

Alternative Method

Using trigonometric ratios, we can express the lengths in terms of

\theta = \angle A

. Since

CD = AB \sin\theta \cos\theta

, we have

CE = CD \sin\theta

and

CF = CD \cos\theta

. The area of triangle

CEF

is

\frac{1}{2} CE \cdot CF = \frac{1}{2} CD^2 \sin\theta \cos\theta = CD^2 \frac{\text{Area}(ABC)}{AB^2}

, which gives the area ratio as

\left(\frac{CD}{AB}\right)^2

. Since

AB = 25

and

CD = 12

, the ratio is

\left(\frac{12}{25}\right)^2 = \frac{144}{625}

.

Estimated Time:3m 0s

Question 71Question →

In a right triangle, one of the acute angles is $\theta$ . If $\tan(\theta) = \frac{x}{y}$ , where $x$ and $y$ are positive constants, what is the value of $\cos\left(\frac{\pi}{2} - \theta\right)$ in terms of $x$ and $y$ ?

\frac{y}{\sqrt{x^2 + y^2}}

\frac{x}{\sqrt{x^2 + y^2}}

\frac{y}{x}

\frac{x}{x^2 + y^2}

Show answer & explanation

Answer: $\frac{x}{\sqrt{x^2 + y^2}}$

Answer

\frac{x}{\sqrt{x^2 + y^2}}

Using the cofunction identity,

\cos\left(\frac{\pi}{2} - \theta\right)

is equivalent to

\sin(\theta)

. For an angle

\theta

in a right triangle,

\tan(\theta) = \frac{x}{y}

defines the ratio of the opposite side (

x

) to the adjacent side (

y

). Applying the Pythagorean theorem, the hypotenuse is

\sqrt{x^2 + y^2}

. Therefore,

\sin(\theta)

, which is the ratio of the opposite side to the hypotenuse, is equal to

\frac{x}{\sqrt{x^2 + y^2}}

.

Step-by-Step Solution

1

Apply the cofunction identity to rewrite the target expression.

\cos\left(\frac{\pi}{2} - \theta\right) = \sin(\theta)

By the cofunction identities, the sine of an acute angle is equal to the cosine of its complementary angle.

2

Relate the given tangent ratio to the side lengths of a right triangle.

\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{y}

, so the opposite side can be represented as

x

and the adjacent side as

y

.

The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.

3

Apply the Pythagorean theorem to calculate the hypotenuse of the triangle.

\text{Hypotenuse} = \sqrt{x^2 + y^2}

The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

4

Express the sine of the angle as a ratio of the opposite side to the hypotenuse.

\sin(\theta) = \frac{x}{\sqrt{x^2 + y^2}}

The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse.

Key Concept

Cofunction identities and right triangle trigonometric ratios

Estimated Time:1m 30s

Question 72Question →

In the figure, triangle $RST$ is a right triangle with the right angle at vertex $S$ . If the length of side $RS$ is $8$ and the length of side $ST$ is $15$ , what is the length of the hypotenuse $RT$ ?

7

13

17

23

Show answer & explanation

Answer: 17

Answer

17

According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the legs:

RT^2 = RS^2 + ST^2

. Substituting the given values yields

RT^2 = 8^2 + 15^2 = 64 + 225 = 289

. Taking the square root of

289

gives

RT = 17

.

Step-by-Step Solution

1

Set up the Pythagorean theorem for right triangle

RST

.

RS^2 + ST^2 = RT^2

In any right triangle, the sum of the squares of the legs equals the square of the hypotenuse.

2

Substitute the given side lengths into the equation.

8^2 + 15^2 = RT^2

The legs of the triangle are

RS = 8

and

ST = 15

.

3

Calculate the squares and sum them.

64 + 225 = 289 = RT^2

8^2 = 64

and

15^2 = 225

. Adding them gives

289

.

4

Take the square root of both sides to find the length of the hypotenuse.

RT = \sqrt{289} = 17

The length must be a positive value, and

\sqrt{289}

simplifies to

17

.

Key Concept

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs:

a^2 + b^2 = c^2

.

Estimated Time:45s

Question 73Question →

A wheel rotates through a central angle of $\frac{11\pi}{18}$ radians. What is the measure of this angle in degrees?

Show answer & explanation

Answer: 110

Answer

110

To convert radians to degrees, multiply the angle by

\frac{180}{\pi}

. Thus,

\frac{11\pi}{18} \times \frac{180}{\pi} = 110

. The measure of the angle is 110 degrees.

Step-by-Step Solution

1

Identify the conversion relationship between radians and degrees.

Multiply the angle in radians by

\frac{180}{\pi}

to convert to degrees.

Since

\pi

radians is equal to

180

degrees, the conversion factor is

\frac{180}{\pi}

.

2

Multiply the given radian measure of

\frac{11\pi}{18}

by the conversion factor.

110

Applying the conversion factor simplifies the expression by canceling

\pi

and dividing

180

by

18

to get

10

, which is then multiplied by

11

to get the final degree measure.

Key Concept

To convert an angle from radians to degrees, multiply the angle in radians by

\frac{180}{\pi}

.

Estimated Time:45s

Question 74Question →

In triangle $ABC$ , the angle at vertex $B$ is a right angle. The lengths of sides $AB$ and $BC$ are $12$ and $16$ , respectively. A point $D$ is chosen on the hypotenuse $AC$ such that $AD = 5$ . A line drawn through $D$ perpendicular to $AC$ intersects the line passing through $B$ and $C$ at point $G$ , such that $B$ lies between $G$ and $C$ . What is the length of segment $GD$ ?

Show answer & explanation

Answer: 11.25

Answer

The length of segment

GD

is 11.25.

By the Pythagorean theorem, the hypotenuse

AC

of right triangle

ABC

is

\sqrt{12^2 + 16^2} = 20

. Subtracting the length of

AD

from

AC

gives

DC = 20 - 5 = 15

. Because the line

GD

is perpendicular to

AC

, the angle

\angle GDC

is

90^\circ

. The triangles

\triangle GDC

and

\triangle ABC

share the angle at vertex

C

and both have a right angle, which means they are similar by Angle-Angle (AA) similarity:

\triangle GDC \sim \triangle ABC

. Using the ratio of corresponding sides, we have

\frac{GD}{AB} = \frac{DC}{BC}

, which translates to

\frac{GD}{12} = \frac{15}{16}

. Solving for

GD

yields

GD = 12 \times \frac{15}{16} = 11.25

.

Step-by-Step Solution

1

Calculate the length of the hypotenuse

AC

using the Pythagorean theorem in right triangle

ABC

.

AC = \sqrt{12^2 + 16^2} = 20

The length of

AC

is required to find the segment lengths on the hypotenuse.

2

Determine the length of segment

DC

.

DC = AC - AD = 20 - 5 = 15

The segment

DC

is a side of the similar triangle

\triangle GDC

that corresponds to side

BC

in

\triangle ABC

.

3

Establish the similarity between triangles

\triangle GDC

and

\triangle ABC

.

\triangle GDC \sim \triangle ABC

by AA similarity

Both triangles share the angle at vertex

C

, and both have a right angle (

\angle GDC = \angle ABC = 90^\circ

).

4

Set up the ratio of corresponding sides and solve for

GD

.

\frac{GD}{AB} = \frac{DC}{BC} \Rightarrow GD = 12 \times \frac{15}{16} = 11.25

The ratio of corresponding sides in similar triangles is equal.

Key Concept

Using right triangle similarity and the Pythagorean theorem to solve for unknown side lengths.

Question 75Question →

If the measure of an angle is $\frac{5\pi}{4}$ radians, what is its measure in degrees?

144^\circ

200^\circ

225^\circ

450^\circ

Show answer & explanation

Answer: $225^\circ$

Answer

225^\circ

To convert an angle from radians to degrees, multiply the radian measure by

\frac{180^\circ}{\pi}

. Multiplying

\frac{5\pi}{4}

by

\frac{180^\circ}{\pi}

simplifies to

\frac{5}{4} \times 180^\circ = 5 \times 45^\circ = 225^\circ

. Therefore, the correct degree measure of the angle is

225^\circ

.

Step-by-Step Solution

1

Set up the conversion from radians to degrees by multiplying the given radian measure by

\frac{180^\circ}{\pi}

.

\frac{5\pi}{4} \times \frac{180^\circ}{\pi}

Since

\pi

radians is equal to

180^\circ

, multiplying by the conversion factor

\frac{180^\circ}{\pi}

converts the units from radians to degrees.

2

Simplify the expression by canceling the

\pi

terms and performing the arithmetic.

225^\circ

The

\pi

in the numerator and denominator cancel out, leaving

\frac{5}{4} \times 180^\circ

. Dividing

180

by

4

gives

45

, and multiplying

5

by

45

yields

225

.

Key Concept

To convert an angle from radians to degrees, multiply the angle in radians by

\frac{180^\circ}{\pi}

.

Alternative Method

Another way to convert is to substitute

180^\circ

directly for

\pi

in the expression:

\frac{5\pi}{4} = \frac{5(180^\circ)}{4} = 5 \times 45^\circ = 225^\circ

.

Estimated Time:45s

Question 76Question →

For an acute angle $\theta$ , $\cos(\theta) = \frac{5}{13}$ . What is the value of $5\tan(\theta) + 13\sin(\theta)$ ?

Show answer & explanation

Answer: 24

Answer

24

Using the Pythagorean identity

\sin^2(\theta) + \cos^2(\theta) = 1

with

\cos(\theta) = \frac{5}{13}

gives

\sin(\theta) = \frac{12}{13}

because

\theta

is an acute angle. The quotient identity gives

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{12}{5}

. Substituting these ratios into the expression

5\tan(\theta) + 13\sin(\theta)

gives

5\left(\frac{12}{5}\right) + 13\left(\frac{12}{13}\right) = 12 + 12 = 24

.

Step-by-Step Solution

1

Find the value of

\sin(\theta)

using the Pythagorean identity.

\sin(\theta) = \frac{12}{13}

Since

\sin^2(\theta) + \cos^2(\theta) = 1

and

\theta

is an acute angle, the sine value is positive:

\sin(\theta) = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \frac{12}{13}

.

2

Find the value of

\tan(\theta)

using the quotient identity.

\tan(\theta) = \frac{12}{5}

By definition,

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

. Substituting the known values yields

\tan(\theta) = \frac{12/13}{5/13} = \frac{12}{5}

.

3

Substitute the trigonometric ratios into the given expression and simplify.

24

Substituting the values of

\tan(\theta)

and

\sin(\theta)

into

5\tan(\theta) + 13\sin(\theta)

gives

5\left(\frac{12}{5}\right) + 13\left(\frac{12}{13}\right) = 12 + 12 = 24

.

Key Concept

Trigonometric ratios and identities, specifically the Pythagorean identity

\sin^2(\theta) + \cos^2(\theta) = 1

and the definition of tangent as

\frac{\sin(\theta)}{\cos(\theta)}

.

Alternative Method

Alternatively, draw a right triangle with an acute angle

\theta

. Since

\cos(\theta) = \frac{5}{13} = \frac{\text{adjacent}}{\text{hypotenuse}}

, label the adjacent side as 5 and the hypotenuse as 13. By the Pythagorean theorem, the opposite side is

\sqrt{13^2 - 5^2} = 12

. From this triangle,

\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{13}

and

\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{5}

. Substituting these values into the expression gives

5\left(\frac{12}{5}\right) + 13\left(\frac{12}{13}\right) = 24

.

Estimated Time:1m 30s

Question 77Question →

For an acute angle $\theta$ , the sum of its sine and cosine is $\frac{\sqrt{7}}{2}$ . What is the value of $\sin^3(\theta) + \cos^3(\theta)$ ?

\frac{5\sqrt{7}}{4}

\frac{11\sqrt{7}}{16}

\frac{5\sqrt{7}}{16}

\frac{7\sqrt{7}}{8}

Show answer & explanation

Answer: $\frac{5\sqrt{7}}{16}$

Answer

The correct answer is

\frac{5\sqrt{7}}{16}

.

The correct answer is

\frac{5\sqrt{7}}{16}

. Squaring the given equation

\sin(\theta) + \cos(\theta) = \frac{\sqrt{7}}{2}

gives

\sin^2(\theta) + 2\sin(\theta)\cos(\theta) + \cos^2(\theta) = \frac{7}{4}

. Substituting the Pythagorean identity

\sin^2(\theta) + \cos^2(\theta) = 1

gives

1 + 2\sin(\theta)\cos(\theta) = \frac{7}{4}

, which simplifies to

\sin(\theta)\cos(\theta) = \frac{3}{8}

. Factoring the sum of cubes yields

\sin^3(\theta) + \cos^3(\theta) = (\sin(\theta) + \cos(\theta))(1 - \sin(\theta)\cos(\theta))

. Substituting the values results in

\left(\frac{\sqrt{7}}{2}\right)\left(1 - \frac{3}{8}\right) = \frac{5\sqrt{7}}{16}

.

Step-by-Step Solution

1

Square both sides of the given equation

\sin(\theta) + \cos(\theta) = \frac{\sqrt{7}}{2}

.

(\sin(\theta) + \cos(\theta))^2 = \sin^2(\theta) + 2\sin(\theta)\cos(\theta) + \cos^2(\theta) = \frac{7}{4}

.

This sets up the expression to apply the Pythagorean trigonometric identity.

2

Substitute the Pythagorean identity

\sin^2(\theta) + \cos^2(\theta) = 1

and solve for the product term.

1 + 2\sin(\theta)\cos(\theta) = \frac{7}{4} \implies 2\sin(\theta)\cos(\theta) = \frac{3}{4} \implies \sin(\theta)\cos(\theta) = \frac{3}{8}

.

This determines the value of the product of the sine and cosine ratios.

3

Apply the sum of cubes factorization formula to

\sin^3(\theta) + \cos^3(\theta)

.

\sin^3(\theta) + \cos^3(\theta) = (\sin(\theta) + \cos(\theta))(\sin^2(\theta) - \sin(\theta)\cos(\theta) + \cos^2(\theta)) = (\sin(\theta) + \cos(\theta))(1 - \sin(\theta)\cos(\theta))

.

This expresses the target quantity in terms of known sum and product values.

4

Substitute the values of the sum and product into the factored expression and simplify.

\sin^3(\theta) + \cos^3(\theta) = \left(\frac{\sqrt{7}}{2}\right)\left(1 - \frac{3}{8}\right) = \left(\frac{\sqrt{7}}{2}\right)\left(\frac{5}{8}\right) = \frac{5\sqrt{7}}{16}

.

This yields the final simplified numerical ratio.

Key Concept

Pythagorean Identity and Algebraic Manipulation of Trigonometric Ratios

Question 78Question →

For an acute angle $\theta$ , the equation $2\cos^2(\theta) - 5\sin(\theta) + 1 = 0$ is true. What is the value of $\sin(\theta)$ ?

Show answer & explanation

Answer: 0.5

Answer

The value of

\sin(\theta)

is

0.5

(or

\frac{1}{2}

)

By applying the Pythagorean identity

\cos^2(\theta) = 1 - \sin^2(\theta)

, the equation

2\cos^2(\theta) - 5\sin(\theta) + 1 = 0

can be rewritten entirely in terms of

\sin(\theta)

, yielding

-2\sin^2(\theta) - 5\sin(\theta) + 3 = 0

. Multiplying by

-1

gives the standard quadratic equation

2\sin^2(\theta) + 5\sin(\theta) - 3 = 0

, which factors as

(2\sin(\theta) - 1)(\sin(\theta) + 3) = 0

. Solving for

\sin(\theta)

yields

\sin(\theta) = 0.5

or

\sin(\theta) = -3

. Since the sine value of any angle must be in the range

[-1, 1]

and the sine of an acute angle must be positive,

\sin(\theta) = 0.5

is the only valid solution.

Step-by-Step Solution

1

Apply the Pythagorean identity to rewrite the cosine term.

2(1 - \sin^2(\theta)) - 5\sin(\theta) + 1 = 0

The equation contains both

\cos^2(\theta)

and

\sin(\theta)

. Substituting

\cos^2(\theta) = 1 - \sin^2(\theta)

allows the equation to be expressed in terms of a single trigonometric function,

\sin(\theta)

.

2

Distribute and simplify the equation into standard quadratic form.

2\sin^2(\theta) + 5\sin(\theta) - 3 = 0

Expanding the equation yields

2 - 2\sin^2(\theta) - 5\sin(\theta) + 1 = 0

, which simplifies to

-2\sin^2(\theta) - 5\sin(\theta) + 3 = 0

. Multiplying the entire equation by

-1

puts it into standard quadratic form

as^2 + bs + c = 0

.

3

Factor the quadratic expression.

(2\sin(\theta) - 1)(\sin(\theta) + 3) = 0

Finding two numbers that multiply to

-6

(from

2 \times -3

) and add to

5

leads to the factors

6

and

-1

. Splitting the middle term and factoring by grouping yields

(2\sin(\theta) - 1)(\sin(\theta) + 3) = 0

.

4

Determine the valid solution based on the angle's constraints.

\sin(\theta) = 0.5

Setting each factor to zero gives

\sin(\theta) = 0.5

or

\sin(\theta) = -3

. Since the sine of any real angle must be between

-1

and

1

,

\sin(\theta) = -3

is undefined. Furthermore, because

\theta

is an acute angle (

0^\circ < \theta < 90^\circ

), the sine value must be positive, confirming

\sin(\theta) = 0.5

.

Key Concept

Pythagorean identity and quadratic trigonometric equations

Estimated Time:2m 0s

Question 79Question →

For an acute angle $x$ measured in degrees, $\sin(x) - \sin(90^\circ - x) = \frac{1}{5}$ . What is the value of $12(\tan(x) + \tan(90^\circ - x))$ ?

Show answer & explanation

Answer: 25

Answer

The value of the expression is 25.

Applying the co-function identity

\sin(90^\circ - x) = \cos(x)

allows the given equation to be written as

\sin(x) - \cos(x) = \frac{1}{5}

. Squaring both sides of this equation yields

\sin^2(x) - 2\sin(x)\cos(x) + \cos^2(x) = \frac{1}{25}

. Applying the Pythagorean identity

\sin^2(x) + \cos^2(x) = 1

simplifies this to

1 - 2\sin(x)\cos(x) = \frac{1}{25}

, which gives

\sin(x)\cos(x) = \frac{12}{25}

. The expression to be evaluated is

12(\tan(x) + \tan(90^\circ - x))

. Using the identity

\tan(90^\circ - x) = \cot(x)

, this expression can be rewritten as

12\left(\frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)}\right) = 12\left(\frac{\sin^2(x) + \cos^2(x)}{\sin(x)\cos(x)}\right) = 12\left(\frac{1}{\sin(x)\cos(x)}\right)

. Substituting the value of

\sin(x)\cos(x)

gives

12 \times \frac{25}{12} = 25

.

Step-by-Step Solution

1

Apply the co-function identity to rewrite the equation.

\sin(x) - \cos(x) = \frac{1}{5}

Since

\sin(90^\circ - x) = \cos(x)

for any angle

x

, we can substitute

\cos(x)

into the given equation.

2

Square both sides of the rewritten equation.

\sin^2(x) - 2\sin(x)\cos(x) + \cos^2(x) = \frac{1}{25}

Squaring both sides allows us to use the Pythagorean trigonometric identity to find the product of sine and cosine.

3

Substitute the Pythagorean identity and solve for

\sin(x)\cos(x)

.

\sin(x)\cos(x) = \frac{12}{25}

Substituting

\sin^2(x) + \cos^2(x) = 1

yields

1 - 2\sin(x)\cos(x) = \frac{1}{25}

, which simplifies to

\sin(x)\cos(x) = \frac{12}{25}

.

4

Rewrite the target expression in terms of sine and cosine.

\tan(x) + \tan(90^\circ - x) = \frac{1}{\sin(x)\cos(x)}

Using the co-function identity

\tan(90^\circ - x) = \cot(x)

and expressing the tangent and cotangent functions as ratios of sine and cosine yields

\frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)} = \frac{\sin^2(x) + \cos^2(x)}{\sin(x)\cos(x)} = \frac{1}{\sin(x)\cos(x)}

.

5

Substitute the value of

\sin(x)\cos(x)

and multiply by 12.

25

Substituting

\sin(x)\cos(x) = \frac{12}{25}

into

12(\tan(x) + \tan(90^\circ - x))

gives

12 \times \frac{25}{12} = 25

.

Key Concept

Applying co-function identities, the Pythagorean identity, and fundamental trigonometric relations to simplify expressions.

Estimated Time:2m 0s

Question 80Question →

A circle has a radius of $10$ . A shaded sector of the circle has an area of $10\pi$ . What is the measure, in degrees, of the central angle of the shaded sector?

Show answer & explanation

Answer: 36

Answer

The measure of the central angle of the shaded sector is 36 degrees.

The total area of the circle is

A = \pi (10)^2 = 100\pi

. The shaded sector's area is

10\pi

, which represents

\frac{10\pi}{100\pi} = \frac{1}{10}

of the total area of the circle. Since a circle consists of

360^\circ

, the central angle of the sector is

\frac{1}{10} \times 360^\circ = 36^\circ

.

Step-by-Step Solution

1

Calculate the total area of the circle.

Total area =

100\pi

The area of a circle is calculated using the formula

A = \pi r^2

where the radius

r = 10

.

2

Determine the proportion of the circle covered by the shaded sector.

Proportion =

\frac{1}{10}

Dividing the sector's area of

10\pi

by the total area of

100\pi

gives the fraction of the circle represented by the sector.

3

Find the central angle in degrees.

Central angle = 36

Since a full circle has a central angle of

360^\circ

, multiplying the proportion

\frac{1}{10}

by

360^\circ

yields the central angle of the sector.

Key Concept

The ratio of the area of a sector to the total area of a circle is equal to the ratio of the sector's central angle measure to the total degree measure of a circle (

360^\circ

).

Estimated Time:45s

Geometry and Trigonometry

Answer

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