Geometry and Trigonometry Practice Questions — SAT

Question 1Question →

In right triangle $PQR$ , the measure of angle $Q$ is $90^\circ$ . If $\tan(P) = \frac{4}{3}$ , what is the value of $\sin(P)$ ?

\frac{4}{5}

\frac{3}{5}

\frac{3}{4}

\frac{5}{4}

Show answer & explanation

Answer: $\frac{4}{5}$

Answer

\frac{4}{5}

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

\tan(P) = \frac{4}{3}

, we can set the length of the side opposite to angle P as

4

and the adjacent side as

3

. Using the Pythagorean theorem, the hypotenuse is

\sqrt{3^2 + 4^2} = 5

. Since the sine of an angle is the ratio of the opposite side to the hypotenuse,

\sin(P) = \frac{4}{5}

.

Step-by-Step Solution

1

Identify the relationship between the tangent ratio and the sides of the right triangle.

The tangent of angle P is defined as the opposite side divided by the adjacent side. Since

\tan(P) = \frac{4}{3}

, we can set the opposite side to

4

and the adjacent side to

3

.

This allows us to establish the relative lengths of the sides of the right triangle.

2

Calculate the length of the hypotenuse using the Pythagorean theorem.

The hypotenuse is

\sqrt{3^2 + 4^2} = \sqrt{25} = 5

.

The hypotenuse is required to calculate the sine ratio.

3

Determine the sine of angle P.

The sine of angle P is defined as the opposite side divided by the hypotenuse, which is

\frac{4}{5}

.

This yields the final requested trigonometric ratio.

Key Concept

Using basic trigonometric ratios (SOH CAH TOA) and the Pythagorean theorem to evaluate trigonometric ratios in a right triangle.

Estimated Time:45s

Question 2Question →

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ . If $\sin(A) - \cos(A) = \frac{1}{5}$ , what is the value of $\sin(A)\sin(B)$ ?

\frac{13}{25}

\frac{16}{25}

\frac{12}{25}

\frac{24}{25}

Show answer & explanation

Answer: $\frac{12}{25}$

Answer

\frac{12}{25}

By the co-function identity for complementary angles in a right triangle,

\sin(B) = \cos(A)

, which means the expression

\sin(A)\sin(B)

is equivalent to

\sin(A)\cos(A)

. Squaring the equation

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

gives

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

. Substituting the Pythagorean identity

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

simplifies this to

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

. Solving for

\sin(A)\cos(A)

yields

2\sin(A)\cos(A) = \frac{24}{25}

, or

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

.

Step-by-Step Solution

1

Express the target expression in terms of a single angle using complementary relationships.

Since angle

C = 90^\circ

, angles

A

and

B

are complementary, so

\sin(B) = \cos(A)

. Thus,

\sin(A)\sin(B) = \sin(A)\cos(A)

.

In any right triangle, the sine of one acute angle equals the cosine of the other acute angle.

2

Square both sides of the given equation

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

.

(\sin(A) - \cos(A))^2 = \left(\frac{1}{5}\right)^2 \implies \sin^2(A) - 2\sin(A)\cos(A) + \cos^2(A) = \frac{1}{25}

.

Squaring the difference allows us to introduce the product term

\sin(A)\cos(A)

and the squared terms.

3

Apply the Pythagorean identity to simplify the equation.

Since

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

, the equation becomes

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

.

The sum of the squares of sine and cosine of the same angle is always equal to 1.

4

Solve for the product

\sin(A)\cos(A)

.

2\sin(A)\cos(A) = 1 - \frac{1}{25} = \frac{24}{25} \implies \sin(A)\cos(A) = \frac{12}{25}

.

Subtracting

1/25

from

1

and dividing the resulting fraction by

2

isolates the target product.

Key Concept

Pythagorean identity and co-function relationships in right triangles

Question 3Question →

In right triangle $ABC$ , the measure of angle $B$ is $90^\circ$ . The acute angles $A$ and $C$ satisfy $\sin(A) = \frac{k}{3}$ and $\cos(C) = \frac{4}{k+4}$ for some positive constant $k$ . What is the value of $k$ ?

6

8

2

12

Show answer & explanation

Answer: 2

Answer

2

In right triangle

ABC

, the measure of angle

B

is

90^\circ

, which means the acute angles

A

and

C

are complementary. Therefore, the co-function identity states that

\sin(A) = \cos(C)

. Equating the given expressions yields

\frac{k}{3} = \frac{4}{k+4}

. Cross-multiplying results in

k(k+4) = 12

, which simplifies to

k^2 + 4k - 12 = 0

. Factoring this equation gives

(k+6)(k-2) = 0

. This quadratic has solutions

k = -6

and

k = 2

. Since

k

must be positive, the value of

k

is

2

.

Step-by-Step Solution

1

Identify the relationship between the acute angles in a right triangle.

Since the measure of angle

B

is

90^\circ

, the sum of angles

A

and

C

is

90^\circ

, which means they are complementary angles. By the co-function identity,

\sin(A) = \cos(C)

.

In a right triangle, the sine of one acute angle equals the cosine of the other acute angle.

2

Equate the expressions for

\sin(A)

and

\cos(C)

and set up the equation for

k

.

\frac{k}{3} = \frac{4}{k+4}

This sets up the algebraic relation to solve for

k

using the given trigonometric expressions.

3

Solve the algebraic equation for

k

.

Cross-multiplying gives

k(k+4) = 12

, which expands to

k^2 + 4k = 12

. Subtracting

12

from both sides results in the quadratic equation

k^2 + 4k - 12 = 0

. Factoring this equation yields

(k+6)(k-2) = 0

.

Cross-multiplication eliminates the denominators, converting the equation to a quadratic form that can be solved by factoring.

4

Determine the valid positive value for

k

.

The solutions to the equation

(k+6)(k-2) = 0

are

k = -6

and

k = 2

. Since

k

must be a positive constant, we select

k = 2

.

The problem specifies that

k

is a positive constant, so the negative solution is discarded.

Key Concept

Co-function identity relating sine and cosine of complementary angles

Estimated Time:1m 30s

Question 4Question →

Lines $L_1$ and $L_2$ are parallel. Points $P$ and $Q$ lie on line $L_1$ , and points $R$ and $S$ lie on line $L_2$ . Segment $PS$ and segment $QR$ intersect at point $T$ . If the measure of angle $TPQ$ is $38^\circ$ and the measure of angle $PTQ$ is $105^\circ$ , what is the measure, in degrees, of angle $TRS$ ?

Show answer & explanation

Answer: 37

Answer

37

To find the measure of angle

TRS

, we first find the measure of angle

TQP

using the property that the sum of interior angles in triangle

TPQ

is

180^\circ

. This gives

\angle TQP = 180^\circ - 38^\circ - 105^\circ = 37^\circ

. Since lines

L_1

and

L_2

are parallel, the transversal line

QR

creates alternate interior angles that are equal in measure. Therefore, angle

TRS

is equal to angle

TQP

, which is

37^\circ

.

Step-by-Step Solution

1

Calculate the measure of angle

TQP

in triangle

TPQ

.

\angle TQP = 37^\circ

The sum of the angles in a triangle is

180^\circ

. Therefore,

\angle TQP = 180^\circ - \angle TPQ - \angle PTQ = 180^\circ - 38^\circ - 105^\circ = 37^\circ

.

2

Relate angle

TQP

to angle

TRS

using parallel line properties.

\angle TRS = 37^\circ

Since line

L_1

is parallel to line

L_2

, the transversal line

QR

creates equal alternate interior angles. Thus,

\angle TRS = \angle TQP = 37^\circ

.

Key Concept

Alternate interior angles and triangle angle sum theorem

Question 5Question →

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ . If $\cos(A) = k \sin(A)$ for some positive constant $k$ , which of the following expressions represents $\cos(B)$ in terms of $k$ ?

\frac{k}{\sqrt{1 + k^2}}

\frac{1}{\sqrt{1 - k^2}}

\frac{1}{\sqrt{1 + k^2}}

\sqrt{1 + k^2}

Show answer & explanation

Answer: $\frac{1}{\sqrt{1 + k^2}}$

Answer

The expression that represents

\cos(B)

is

\frac{1}{\sqrt{1 + k^2}}

.

In right triangle

ABC

with right angle

C

, the acute angles

A

and

B

are complementary (

A + B = 90^\circ

). By the co-function identity,

\cos(B) = \sin(A)

. Using the given relation

\cos(A) = k \sin(A)

and substituting it into the Pythagorean identity

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

, we get

\sin^2(A) + (k \sin(A))^2 = 1 \implies \sin^2(A)(1 + k^2) = 1

. Solving for

\sin(A)

gives

\sin(A) = \frac{1}{\sqrt{1 + k^2}}

because

\sin(A)

must be positive for an acute angle. Since

\cos(B) = \sin(A)

, we conclude that the correct expression is

\frac{1}{\sqrt{1 + k^2}}

.

Step-by-Step Solution

1

Use the relationship between the acute angles in right triangle

ABC

.

Since angle

C

is

90^\circ

, angles

A

and

B

are complementary. Thus,

\cos(B) = \sin(A)

.

This allows us to convert the target term

\cos(B)

into

\sin(A)

, which can be related to the given equation.

2

Substitute the given relation

\cos(A) = k \sin(A)

into the Pythagorean identity.

Using

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

, we substitute to get

\sin^2(A) + (k \sin(A))^2 = 1

, which simplifies to

\sin^2(A)(1 + k^2) = 1

.

This sets up a single equation containing only

\sin(A)

and the constant

k

.

3

Solve for

\sin(A)

and substitute back to find

\cos(B)

.

Solving for

\sin(A)

gives

\sin(A) = \frac{1}{\sqrt{1 + k^2}}

(since

\sin(A) > 0

for acute angle

A

). Since

\cos(B) = \sin(A)

, we have

\cos(B) = \frac{1}{\sqrt{1 + k^2}}

.

This yields the final value of

\cos(B)

in terms of

k

.

Key Concept

Applying complementary angle trigonometric identities (co-functions) and the Pythagorean identity in a right triangle.

Alternative Method

Alternatively, you can model this by setting up a right triangle. Since

\cos(A) = k \sin(A)

, we can divide both sides by

\sin(A)

to get

\cot(A) = k

, which means

\tan(A) = \frac{1}{k}

. In a right triangle

ABC

with right angle

C

,

\tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AC} = \frac{1}{k}

. Let

BC = 1

and

AC = k

. By the Pythagorean theorem, the hypotenuse

AB = \sqrt{1^2 + k^2} = \sqrt{1 + k^2}

. Then,

\cos(B) = \frac{\text{adjacent to } B}{\text{hypotenuse}} = \frac{BC}{AB} = \frac{1}{\sqrt{1 + k^2}}

.

Estimated Time:2m 0s

Question 6Question →

A circle in the $xy$ -plane is defined by the equation $(x - 7)^2 + (y + 2)^2 = 36$ . What is the diameter of this circle?

6

12

36

72

Show answer & explanation

Answer: 12

Answer

12

The standard equation of a circle is

(x - h)^2 + (y - k)^2 = r^2

, where

(h, k)

is the center of the circle and

r

is the radius. In the equation

(x - 7)^2 + (y + 2)^2 = 36

, the value of

r^2

is 36. Taking the square root gives the radius,

r = 6

. The diameter of a circle is twice its radius, so the diameter is

2(6) = 12

.

Step-by-Step Solution

1

Identify the value of the radius squared,

r^2

, from the given equation of the circle.

r^2 = 36

In the standard equation of a circle,

(x - h)^2 + (y - k)^2 = r^2

, the constant on the right side of the equation is equal to the square of the radius.

2

Determine the radius,

r

, by taking the square root of

r^2

.

r = \sqrt{36} = 6

Since the radius represents a distance, it must be the positive square root of the constant term.

3

Calculate the diameter of the circle by multiplying the radius by 2.

Diameter =

2 \times 6 = 12

The diameter of a circle is defined as twice its radius.

Key Concept

Identifying circle properties from its standard equation form

(x - h)^2 + (y - k)^2 = r^2

.

Question 7Question →

A robotic arm rotates counterclockwise by $210^\circ$ and then rotates clockwise by $75^\circ$ . What is the net counterclockwise rotation of the robotic arm, in radians?

\frac{3\pi}{4}

\frac{4\pi}{3}

\frac{29\pi}{36}

\frac{3\pi}{5}

Show answer & explanation

Answer: $\frac{3\pi}{4}$

Answer

The correct answer is

\frac{3\pi}{4}

radians.

The correct answer is determined by first finding the net angle of rotation. Subtracting the

75^\circ

clockwise rotation from the

210^\circ

counterclockwise rotation yields a net rotation of

135^\circ

counterclockwise. Converting this angle to radians is done by multiplying

135^\circ

by

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

, resulting in

\frac{135\pi}{180}

radians. Dividing both the numerator and denominator by 45 simplifies the fraction to

\frac{3\pi}{4}

radians.

Step-by-Step Solution

1

Calculate the net counterclockwise rotation in degrees.

210^\circ - 75^\circ = 135^\circ

Since counterclockwise rotation is positive and clockwise rotation is negative, we subtract the clockwise rotation from the counterclockwise rotation.

2

Convert the angle from degrees to radians.

135 \times \frac{\pi}{180} = \frac{135\pi}{180}

radians

To convert degrees to radians, multiply the degree measure by the conversion factor

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

.

3

Simplify the fraction to its lowest terms.

\frac{3\pi}{4}

radians

Divide both the numerator and the denominator by their greatest common divisor, which is 45.

Key Concept

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

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

and simplify the resulting fraction.

Alternative Method

Alternatively, convert each individual rotation to radians first, then subtract. Convert

210^\circ

to radians:

210 \times \frac{\pi}{180} = \frac{7\pi}{6}

radians. Convert

75^\circ

to radians:

75 \times \frac{\pi}{180} = \frac{5\pi}{12}

radians. Subtract the clockwise rotation from the counterclockwise rotation:

\frac{7\pi}{6} - \frac{5\pi}{12} = \frac{14\pi}{12} - \frac{5\pi}{12} = \frac{9\pi}{12} = \frac{3\pi}{4}

radians.

Estimated Time:1m 30s

Question 8Question →

In the $xy$ -plane, a right triangle has vertices at the origin $O(0, 0)$ , $A(x, 0)$ , and $B(0, y)$ , where $x > 0$ and $y > 0$ . If the length of the hypotenuse $AB$ is $26$ and the slope of the line containing segment $AB$ is $-\frac{12}{5}$ , what is the value of $y$ ?

Show answer & explanation

Answer: 24

Answer

The value of

y

is 24.

The correct answer is 24 because the slope of the line through

A(x, 0)

and

B(0, y)

is

-\frac{y}{x}

. Setting this equal to

-\frac{12}{5}

gives the relationship

y = \frac{12}{5}x

. Substituting this into the Pythagorean theorem equation

x^2 + y^2 = 26^2

yields

x^2 + \left(\frac{12}{5}x\right)^2 = 676

, which simplifies to

\frac{169}{25}x^2 = 676

. Solving for

x

gives

x = 10

, and multiplying by

\frac{12}{5}

gives the value of

y

as 24.

Step-by-Step Solution

1

Express the slope of the line containing segment

AB

in terms of

x

and

y

.

y = \frac{12}{5}x

The line passes through

A(x, 0)

and

B(0, y)

, so its slope is

-\frac{y}{x}

. Setting this equal to the given slope of

-\frac{12}{5}

gives the relationship between

x

and

y

.

2

Apply the Pythagorean theorem to the right triangle

OAB

.

x^2 + y^2 = 676

The lengths of the legs of the right triangle are

x

and

y

, and the hypotenuse is

26

.

3

Substitute the slope relationship into the Pythagorean equation to solve for

x

.

x = 10

Substituting

y = \frac{12}{5}x

gives

x^2 + \frac{144}{25}x^2 = 676

, which simplifies to

\frac{169}{25}x^2 = 676

. Solving for

x

gives

10

.

4

Calculate the value of

y

.

y = 24

Using

x = 10

in

y = \frac{12}{5}x

gives

24

.

Key Concept

Solving right triangle problems in the coordinate plane by combining linear equations (slope) with the Pythagorean theorem.

Question 9Question →

A sign is in the shape of a parallelogram with a base of $18$ inches and a height of $10$ inches. A smaller, similar parallelogram is cut out from the center of the sign. If the scale factor of the smaller parallelogram to the larger parallelogram is $1$ to $3$ , what is the area, in square inches, of the remaining portion of the sign?

Show answer & explanation

Answer: 160

Answer

160

The area of the larger parallelogram is

18 \times 10 = 180

square inches. Since the smaller parallelogram is similar to the larger one with a scale factor of

1

to

3

, the area of the smaller parallelogram is

\left(\frac{1}{3}\right)^2 = \frac{1}{9}

of the area of the larger parallelogram. This gives an area of

180 \times \frac{1}{9} = 20

square inches for the smaller parallelogram. Subtracting this from the larger area yields

180 - 20 = 160

square inches.

Step-by-Step Solution

1

Calculate the area of the larger parallelogram.

A_{\text{larger}} = 180

square inches

The area of a parallelogram is the product of its base and height (

A = bh

).

2

Determine the area of the smaller, similar parallelogram.

A_{\text{smaller}} = 20

square inches

The ratio of the areas of similar figures is the square of the scale factor:

\left(\frac{1}{3}\right)^2 = \frac{1}{9}

.

3

Subtract the smaller area from the larger area to find the remaining area.

A_{\text{remaining}} = 160

square inches

The remaining portion of the sign is the total area minus the cutout area.

Key Concept

The area of similar geometric shapes scales by the square of the linear scale factor.

Question 10Question →

A circle with center $O$ has a radius of 15. Chord $AB$ of the circle has a length of 18, and chord $CD$ is parallel to $AB$ and has a length of 24. If the two chords are on opposite sides of the center $O$ , what is the distance between chord $AB$ and chord $CD$ ?

Show answer & explanation

Answer: 21

Answer

21

The distance between the chords is 21. By drawing a line through the center

O

perpendicular to both chords, we bisect chord

AB

at

M

(length 9) and chord

CD

at

N

(length 12). Using the Pythagorean theorem with the radius of 15, the perpendicular distance from the center to chord

AB

is

\sqrt{15^2 - 9^2} = 12

and to chord

CD

is

\sqrt{15^2 - 12^2} = 9

. Since the chords lie on opposite sides of the center, the total distance between them is the sum of these distances,

12 + 9 = 21

.

Step-by-Step Solution

1

Find the perpendicular distance from the center

O

to chord

AB

.

OM = 12

The perpendicular from the center bisects the chord, so we use the Pythagorean theorem in right triangle

OMA

with hypotenuse

OA = 15

and leg

AM = 9

.

2

Find the perpendicular distance from the center

O

to chord

CD

.

ON = 9

Similarly, we use the Pythagorean theorem in right triangle

ONC

with hypotenuse

OC = 15

and leg

CN = 12

.

3

Add the perpendicular distances together to find the total distance between the chords.

21

Because the chords are parallel and on opposite sides of the center, the total distance between them is the sum of their individual distances to the center.

Key Concept

The perpendicular from the center of a circle to a chord bisects the chord, and the distance from the center to the chord can be calculated using the Pythagorean theorem with the radius of the circle.

Question 11Question →

A surveyor stands at point $A$ on one bank of a straight river, directly across from a tree $T$ on the opposite bank. She walks along the riverbank to point $B$ such that the line segment $AB$ is perpendicular to $AT$ . The distance from $A$ to $B$ is $7$ meters. She then continues walking in the same direction along the riverbank for another $25$ meters to point $C$ . If the straight-line distance from point $C$ to the tree $T$ is $40$ meters, what is the straight-line distance, in meters, from point $B$ to the tree $T$ ?

8.75

11.2

25

31

Show answer & explanation

Answer: 25

Answer

25

The correct answer is 25. Since the segments

AB

and

BC

lie on a straight line and are perpendicular to

AT

, triangle

TAC

is a right triangle with leg

AC = 7 + 25 = 32

meters and hypotenuse

TC = 40

meters. By the Pythagorean theorem, the common leg

TA

is equal to

\sqrt{40^2 - 32^2} = \sqrt{576} = 24

meters. Using the Pythagorean theorem again on the smaller right triangle

TAB

with legs

TA = 24

meters and

AB = 7

meters, the hypotenuse

TB

is equal to

\sqrt{24^2 + 7^2} = \sqrt{625} = 25

meters.

Step-by-Step Solution

1

Determine the total distance from point

A

to point

C

along the riverbank.

AC = AB + BC = 7 + 25 = 32

meters.

Points

A

,

B

, and

C

lie on a straight line in that order, so their distances add linearly.

2

Apply the Pythagorean theorem to the right triangle

TAC

to find the distance from

A

to the tree

T

.

TA = \sqrt{TC^2 - AC^2} = \sqrt{40^2 - 32^2} = \sqrt{1600 - 1024} = \sqrt{576} = 24

meters.

Since

AT

is perpendicular to

AC

, triangle

TAC

is a right triangle with hypotenuse

TC

and legs

TA

and

AC

.

3

Apply the Pythagorean theorem to the right triangle

TAB

to find the distance from

B

to the tree

T

.

TB = \sqrt{TA^2 + AB^2} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25

meters.

Triangle

TAB

is a right triangle with legs

TA

and

AB

and hypotenuse

TB

.

Key Concept

Using the Pythagorean theorem in multi-step geometric problems with right triangles sharing a common side.

Estimated Time:1m 30s

Question 12Question →

The equation $x^2 + y^2 - 14x - 8y + 40 = 0$ defines a circle in the $xy$ -plane. If the center of the circle has coordinates $(h, k)$ , what is the value of $h + k$ ?

Show answer & explanation

Answer: 11

Answer

The correct answer is 11.

The equation is rewritten in the standard form

(x - 7)^2 + (y - 4)^2 = 25

by completing the square. The center coordinates

(h, k)

are

(7, 4)

. Thus,

h + k = 7 + 4 = 11

.

Step-by-Step Solution

1

Group terms and complete the square for x and y

The equation becomes

(x - 7)^2 + (y - 4)^2 = 25

.

To convert the general form of the circle's equation into the standard form

(x - h)^2 + (y - k)^2 = r^2

.

2

Identify the coordinates of the center

The center

(h, k)

is

(7, 4)

, so

h = 7

and

k = 4

.

The standard form directly gives the center coordinates as

(h, k)

when written as

(x - h)^2 + (y - k)^2 = r^2

.

3

Calculate the sum of the coordinates

h + k = 7 + 4 = 11

The question asks for the value of

h + k

.

Key Concept

Completing the square to find the standard form of a circle's equation and identifying the center coordinates.

Question 13Question →

In the $xy$ -plane, the graph of the equation $x^2 + y^2 + 8x - 10y = w$ , where $w$ is a constant, is a circle. If the circle is tangent to the $x$ -axis, what is the value of $w$ ?

-16

-25

-36

66

Show answer & explanation

Answer: -16

Answer

-16

Completing the square on the given equation gives

(x+4)^2 + (y-5)^2 = w + 41

. The center of this circle is

(-4, 5)

. If a circle is tangent to the

x

-axis, the radius is equal to the distance from the center to the

x

-axis, which is the absolute value of the

y

-coordinate of the center,

|5| = 5

. Thus, the radius squared is

5^2 = 25

. Equating

w + 41

to

25

gives

w = -16

.

Step-by-Step Solution

1

Complete the square for the

x

and

y

terms in the equation

x^2 + y^2 + 8x - 10y = w

.

(x+4)^2 + (y-5)^2 = w + 41

Completing the square allows us to find the center

(h, k)

and the radius squared

r^2

of the circle in the standard form

(x-h)^2 + (y-k)^2 = r^2

.

2

Identify the center of the circle and determine the radius based on the tangency to the

x

-axis.

Center is

(-4, 5)

and radius

r = 5

.

The standard form shows the center is

(-4, 5)

. Since the circle is tangent to the

x

-axis, the radius is the vertical distance from the center to the line

y=0

(the

x

-axis), which is

|5| = 5

.

3

Set

r^2 = w + 41

and solve for

w

.

w = -16

Since

r = 5

, we have

r^2 = 25

. Equating

w + 41

to

25

gives

w = 25 - 41 = -16

.

Key Concept

Equations of circles in the coordinate plane can be written in the standard form

(x-h)^2 + (y-k)^2 = r^2

by completing the square. When a circle is tangent to the

x

-axis, its radius is equal to the absolute value of the

y

-coordinate of its center (

|k|

).

Question 14Question →

In the $xy$ -plane, right triangle $ABC$ has vertices at $A(0, 0)$ , $B(k, 0)$ , and $C(0, 6)$ , where $k$ is a positive constant. The right angle of the triangle is at vertex $A$ . If the length of the altitude from $A$ to side $BC$ is $4.8$ , what is the value of $k$ ?

3.6

4.5

8

10

Show answer & explanation

Answer: 8

Answer

8

The correct value is 8. The area of right triangle

ABC

is given by

\frac{1}{2} \times \text{base} \times \text{height}

. Using the legs, the area is

\frac{1}{2} \times k \times 6 = 3k

. Using the hypotenuse and the altitude, the area is

\frac{1}{2} \times BC \times 4.8 = 2.4 \times BC

. Equating the two expressions gives

3k = 2.4 \times BC

, which simplifies to

BC = 1.25k

. Applying the Pythagorean theorem,

(1.25k)^2 = k^2 + 6^2 \implies 1.5625k^2 = k^2 + 36 \implies 0.5625k^2 = 36 \implies k^2 = 64

. Since

k

must be positive,

k = 8

.

Step-by-Step Solution

1

Express the area of right triangle

ABC

in two different ways using its base and height.

Area =

3k

and Area =

2.4 \times BC

.

The area of a right triangle can be calculated using the legs as the base and height, Area =

\frac{1}{2} \times AB \times AC = \frac{1}{2} \times k \times 6 = 3k

. Alternatively, using the hypotenuse

BC

as the base and the altitude to the hypotenuse as the height, Area =

\frac{1}{2} \times BC \times 4.8 = 2.4 \times BC

.

2

Equate the two area expressions to find the length of the hypotenuse

BC

in terms of

k

.

3k = 2.4 \times BC \implies BC = 1.25k

.

Since both expressions represent the area of the same triangle, they must be equal.

3

Apply the Pythagorean theorem to right triangle

ABC

to set up an equation for

k

.

(1.25k)^2 = k^2 + 6^2 \implies 1.5625k^2 = k^2 + 36

.

In right triangle

ABC

, the Pythagorean theorem states that

BC^2 = AB^2 + AC^2

.

4

Solve the equation for

k

.

0.5625k^2 = 36 \implies k^2 = 64 \implies k = 8

(since

k > 0

).

Subtracting

k^2

from both sides gives

0.5625k^2 = 36

. Dividing both sides by

0.5625

yields

k^2 = 64

. Taking the square root gives

k = 8

, since

k

represents a positive coordinate.

Key Concept

Relating the area of a right triangle to its legs and hypotenuse altitude to set up proportions, followed by applying the Pythagorean theorem.

Estimated Time:1m 30s

Question 15Question →

If $\cos(\theta) = \frac{7}{25}$ and $0 < \theta < \frac{\pi}{2}$ , what is the value of $\sin\left(\frac{\pi}{2} - \theta\right)$ ?

\frac{24}{25}

\frac{7}{25}

\frac{25}{7}

\frac{18}{25}

Show answer & explanation

Answer: $\frac{7}{25}$

Answer

\frac{7}{25}

The correct answer is

\frac{7}{25}

because the co-function identity states that for any angle

\theta

, the sine of its complement,

\frac{\pi}{2} - \theta

, is equal to the cosine of

\theta

. Given that

\cos(\theta) = \frac{7}{25}

, it follows that

\sin\left(\frac{\pi}{2} - \theta\right) = \frac{7}{25}

.

Step-by-Step Solution

1

Identify the relevant trigonometric identity.

Use the co-function identity

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

.

Since the angle is given in radians, the complementary angle identity relates the sine of the complement to the cosine of the original angle.

2

Substitute the given value.

Since

\cos(\theta) = \frac{7}{25}

, then

\sin\left(\frac{\pi}{2} - \theta\right) = \frac{7}{25}

.

Direct substitution of the given cosine value into the identity yields the final answer.

Key Concept

Co-function identities relate the sine and cosine of complementary angles.

Question 16Question →

A rectangular prism has a volume of $15\text{ cubic centimeters}$ . If the length, width, and height of the prism are all doubled, what is the volume, in cubic centimeters, of the new rectangular prism?

30

120

60

90

Show answer & explanation

Answer: $120$

Answer

120

The volume of a rectangular prism is proportional to the product of its length, width, and height. Doubling each of these dimensions multiplies the volume by a factor of

2 \times 2 \times 2 = 8

. Multiplying the original volume of

15\text{ cubic centimeters}

by

8

gives

120\text{ cubic centimeters}

.

Step-by-Step Solution

1

Identify how the volume of a rectangular prism relates to its dimensions.

The volume is given by

V = l \times w \times h

.

This establishes the relationship between the linear dimensions and the volume of the solid.

2

Determine the scale factor for the volume when all linear dimensions are doubled.

The new dimensions are

2l

,

2w

, and

2h

, which gives a new volume of

(2l)(2w)(2h) = 8(l \times w \times h) = 8V

. Thus, the volume scale factor is

2^3 = 8

.

Volume scales with the cube of the linear dimensions change.

3

Calculate the volume of the scaled prism using the original volume.

New Volume =

8 \times 15 = 120\text{ cubic centimeters}

.

Multiply the original volume by the volume scale factor to obtain the final answer.

Key Concept

Dimensional scaling of solids (volume scales cubically)

Estimated Time:45s

Question 17Question →

The volume of a right circular cylinder is $72\pi$ . If the radius of the base of the cylinder is $3$ , what is the height of the cylinder?

Show answer & explanation

Answer: 8

Answer

The height of the cylinder is 8.

The volume

V

of a right circular cylinder is given by the formula

V = \pi r^2 h

, where

r

is the base radius and

h

is the height. By substituting the given volume of

72\pi

and radius of

3

, we obtain the equation

72\pi = \pi (3)^2 h

. Simplifying the right side gives

72\pi = 9\pi h

. Dividing both sides of the equation by

9\pi

isolates the height, giving

h = 8

.

Step-by-Step Solution

1

Use the formula for the volume of a right circular cylinder.

V = \pi r^2 h

This formula relates the volume, radius, and height of a cylinder.

2

Substitute the known values into the volume formula.

72\pi = \pi (3)^2 h

To form an equation with the unknown height.

3

Simplify the squared radius and isolate the variable.

h = 8

Simplifying

3^2

gives

9

. Dividing

72\pi

by

9\pi

isolates the height.

Key Concept

Solving for an unknown dimension using the volume formula of a cylinder.

Estimated Time:45s

Question 18Question →

A right circular cylinder has a volume of $20\text{ cubic inches}$ . If the radius of the cylinder is tripled and its height remains the same, what is the volume, in cubic inches, of the new cylinder?

60

120

180

540

Show answer & explanation

Answer: 180

Answer

180

The volume of a right circular cylinder is calculated using the formula

V = \pi r^2 h

. When the radius

r

is tripled to

3r

and the height

h

remains constant, the volume is multiplied by

3^2 = 9

. Thus, the new volume is

9

times the original volume of

20

cubic inches, which equals

180

cubic inches.

Step-by-Step Solution

1

Write down the formula for the volume of a right circular cylinder.

V = \pi r^2 h

To identify how the volume of a cylinder relates to its radius and height.

2

Determine the effect of tripling the radius on the volume formula.

V_{\text{new}} = \pi (3r)^2 h = 9\pi r^2 h

Substituting the new radius

3r

into the formula shows that the new volume is 9 times the original volume.

3

Substitute the original volume of

20

cubic inches into the expression for the new volume.

V_{\text{new}} = 9 \times 20 = 180

Since the original volume

\pi r^2 h

equals

20

, we multiply this value by 9 to find the new volume.

Key Concept

Volume of a cylinder and dimensional scaling

Estimated Time:45s

Question 19Question →

A right circular cone has a volume of $24\text{ cubic centimeters}$ . A second right circular cone has a base radius that is $3$ times the base radius of the first cone, and a height that is half the height of the first cone. What is the volume, in cubic centimeters, of the second cone?

36

108

216

324

Show answer & explanation

Answer: 108

Answer

108

The volume of a cone is proportional to the square of its radius and directly proportional to its height. Tripling the radius increases the volume by a factor of

3^2 = 9

. Halving the height reduces the volume by a factor of

\frac{1}{2}

. Combining these changes scales the original volume by

9 \times \frac{1}{2} = 4.5

. Multiplying the original volume of

24

by

4.5

gives the correct volume of

108

.

Step-by-Step Solution

1

Write down the formula for the volume of a right circular cone.

V = \frac{1}{3}\pi r^2 h

This establishes the relationship between volume, radius, and height.

2

Express the volume of the second cone (

V_2

) in terms of the dimensions of the first cone (

r

and

h

).

V_2 = \frac{1}{3}\pi (3r)^2 \left(\frac{1}{2}h\right)

The radius of the second cone is

3r

and the height is

\frac{1}{2}h

.

3

Simplify the expression for the second cone's volume to find the scale factor relative to the first cone.

V_2 = \frac{1}{3}\pi (9r^2) \left(\frac{1}{2}h\right) = \frac{9}{2} \left(\frac{1}{3}\pi r^2 h\right) = 4.5 V_1

Squaring the radius multiplier of 3 gives a factor of 9, and the height multiplier of 0.5 gives a factor of 0.5, resulting in a combined scaling factor of 4.5.

4

Calculate the final volume using the volume of the first cone (

V_1 = 24

).

V_2 = 4.5 \times 24 = 108

Multiplying the original volume of

24\text{ cm}^3

by the overall scale factor of

4.5

yields the new volume.

Key Concept

Dimensional scaling of three-dimensional solids

Question 20Question →

A right circular cylinder has a height of $12$ and a base radius of $r$ . A sphere has a radius of $r$ . If the volume of the cylinder is equal to the volume of the sphere, what is the value of $r$ ?

Show answer & explanation

Answer: 9

Answer

9

The volume of a cylinder is given by

V = \pi r^2 h

and the volume of a sphere is given by

V = \frac{4}{3}\pi r^3

. Given that the cylinder's height is

12

, its volume is

12\pi r^2

. Setting the volumes equal yields

12\pi r^2 = \frac{4}{3}\pi r^3

. Since

r

is a non-zero radius, we can divide both sides by

\pi r^2

, resulting in

12 = \frac{4}{3}r

. Multiplying both sides by

\frac{3}{4}

gives

r = 9

.

Step-by-Step Solution

1

State the standard volume formulas for a right circular cylinder and a sphere.

V_{\text{cylinder}} = \pi r^2 h

and

V_{\text{sphere}} = \frac{4}{3}\pi r^3

These formulas are needed to express the volumes of both solids in terms of

r

.

2

Equate the volume of the cylinder to the volume of the sphere and substitute the given height of

12

.

12\pi r^2 = \frac{4}{3}\pi r^3

The problem states that the volume of the cylinder is equal to the volume of the sphere.

3

Divide both sides of the equation by

\pi r^2

.

12 = \frac{4}{3}r

Simplifies the equation to a first-degree equation in terms of

r

.

4

Solve for

r

by multiplying both sides of the simplified equation by the reciprocal of

\frac{4}{3}

, which is

\frac{3}{4}

.

r = 9

Isolates the variable

r

to find the correct value.

Key Concept

Equating the volumes of geometric solids to solve for an unknown dimension.

Estimated Time:1m 30s

Geometry and Trigonometry

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