Geometry and Trigonometry Practice Questions — SAT — Page 9

Question 161Question →

A sector of a circle with center $O$ has a central angle of $150^\circ$ and an area of $15\pi$ . What is the length of the minor arc that bounds this sector?

5\pi

\frac{5\pi}{2}

12\pi

30\pi

Show answer & explanation

Answer: $5\pi$

Answer

5\pi

To find the arc length, we first determine the radius of the circle using the sector area. The area of a sector with a central angle of

150^\circ

is

\frac{150}{360} = \frac{5}{12}

of the total circle area. Setting up the equation

15\pi = \frac{5}{12}\pi r^2

allows us to solve for

r^2 = 36

, which gives a radius of

r = 6

. Using the radius, we find the length of the minor arc by taking the same fraction of the total circumference:

\frac{5}{12} \times 2\pi(6) = 5\pi

. This matches the correct option.

Step-by-Step Solution

1

Set up the equation for the area of the sector to solve for the radius

r

.

15\pi = \frac{150}{360} \pi r^2

The area of a sector is given by the formula

A = \frac{\theta}{360} \pi r^2

, where

\theta

is the central angle in degrees.

2

Simplify the fraction and solve for

r^2

and

r

.

15\pi = \frac{5}{12} \pi r^2 \implies 15 = \frac{5}{12} r^2 \implies r^2 = 36 \implies r = 6

Dividing both sides by

\pi

and multiplying by

\frac{12}{5}

isolates

r^2

, and taking the square root gives the radius

r

.

3

Calculate the length of the minor arc using the radius and central angle.

Arc Length

= \frac{150}{360} \times 2\pi(6) = \frac{5}{12} \times 12\pi = 5\pi

The arc length formula is

L = \frac{\theta}{360} \times 2\pi r

, representing the fraction of the total circumference.

Key Concept

The relationship between a circle's sector area, central angle, radius, and arc length.

Alternative Method

Instead of solving for the radius first, note that the ratio of the sector area to the total area is equal to the ratio of the arc length to the total circumference. Since Sector Area

= \frac{1}{2} r L

(where

L

is the arc length and

r

is the radius), we have

15\pi = \frac{1}{2} r L

. Since we also know that the area

15\pi = \frac{5}{12} \pi r^2 \implies r = 6

, substituting this directly into the area-arc relation gives

15\pi = \frac{1}{2} (6) L \implies 15\pi = 3L \implies L = 5\pi

.

Estimated Time:1m 30s

Question 162Question →

In the $xy$ -plane, a circle is defined by the equation $x^2 + y^2 + 6x - 8y = 0$ . Which of the following points lies on the circle?

(2, 4)

(-3, 4)

(3, -4)

(0, 5)

Show answer & explanation

Answer: $(2, 4)$

Answer

The point

(2, 4)

lies on the circle.

Substituting the coordinates of the point

(2, 4)

into the circle's equation

x^2 + y^2 + 6x - 8y = 0

yields

2^2 + 4^2 + 6(2) - 8(4) = 4 + 16 + 12 - 32 = 0

, which is a true statement. Therefore, this point lies on the circle.

Step-by-Step Solution

1

Group the

x

and

y

terms and complete the square for both variables.

The expression

(x^2 + 6x) + (y^2 - 8y) = 0

becomes

(x^2 + 6x + 9) + (y^2 - 8y + 16) = 9 + 16

, which simplifies to

(x + 3)^2 + (y - 4)^2 = 25

.

Converting the general form equation of a circle into the standard form

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

helps identify its center and radius.

2

Determine the center and the radius from the standard form equation.

The center is

(h, k) = (-3, 4)

and the radius is

r = \sqrt{25} = 5

.

Understanding the circle's parameters allows us to analyze which points lie on its perimeter.

3

Substitute the given options into the standard form equation to verify which point lies on the circle.

For

(2, 4)

, we get

(2 + 3)^2 + (4 - 4)^2 = 5^2 + 0^2 = 25

, which is true. For the other points, the equations are not satisfied.

A point lies on a circle if its distance to the center is exactly equal to the radius, meaning its coordinates satisfy the circle's equation.

Key Concept

Equations of Circles in the Coordinate Plane

Estimated Time:1m 30s

Question 163Question →

In the $xy$ -plane, the graph of the equation $x^2 + y^2 - 10x - 24y + 69 = 0$ is a circle. What is the distance between the center of this circle and the origin?

Show answer & explanation

Answer: 13

Answer

The distance between the center of the circle and the origin is 13.

Completing the square on the given equation yields standard form

(x - 5)^2 + (y - 12)^2 = 100

, identifying the center of the circle as

(5, 12)

. The distance from

(5, 12)

to

(0,0)

is calculated using the distance formula:

\sqrt{5^2 + 12^2} = \sqrt{169} = 13

.

Step-by-Step Solution

1

Group the

x

and

y

terms and complete the square for each variable.

(x - 5)^2 + (y - 12)^2 = 100

By rewriting

x^2 - 10x

as

(x - 5)^2 - 25

and

y^2 - 24y

as

(y - 12)^2 - 144

, the equation becomes

(x - 5)^2 - 25 + (y - 12)^2 - 144 + 69 = 0

. Combining the constant terms gives

(x - 5)^2 + (y - 12)^2 - 100 = 0

, which simplifies to standard form.

2

Identify the center of the circle from the standard form

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

.

The center is

(5, 12)

.

Comparing

(x - 5)^2 + (y - 12)^2 = 100

to the standard form shows that

h = 5

and

k = 12

.

3

Use the distance formula to calculate the distance between the center

(5, 12)

and the origin

(0, 0)

.

13

The distance formula is

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

. Substituting the points gives

d = \sqrt{(5 - 0)^2 + (12 - 0)^2} = \sqrt{25 + 144} = \sqrt{169} = 13

.

Key Concept

Equations of Circles in the Coordinate Plane

Question 164Question →

In right triangle $XYZ$ , the measure of angle $Y$ is $90^\circ$ . If the length of side $XY$ is $12$ and the area of the triangle is $30$ , what is the value of $\cos(X)$ ?

\frac{12}{13}

\frac{5}{13}

\frac{5}{12}

\frac{24}{25}

Show answer & explanation

Answer: $\frac{12}{13}$

Answer

\frac{12}{13}

The correct answer is the fraction expressing twelve thirteenths. The area of thirty and the leg length of twelve imply that the other leg has a length of five. Applying the Pythagorean theorem to legs of twelve and five yields a hypotenuse of thirteen. Because cosine is defined as the ratio of the adjacent leg to the hypotenuse, the cosine of angle

X

is twelve over thirteen.

Step-by-Step Solution

1

Find the length of leg

YZ

using the area formula.

YZ = 5

The area of a right triangle is

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

. Here,

\text{Area} = 30

and base

XY = 12

, so

30 = \frac{1}{2} \times 12 \times YZ \implies 30 = 6 \times YZ \implies YZ = 5

.

2

Calculate the length of the hypotenuse

XZ

using the Pythagorean theorem.

XZ = 13

In right triangle

XYZ

, the Pythagorean theorem states that

XZ^2 = XY^2 + YZ^2

. Substituting the known leg lengths,

XZ^2 = 12^2 + 5^2 = 144 + 25 = 169

. Taking the square root gives

XZ = 13

.

3

Determine the value of

\cos(X)

using the trigonometric definition.

\cos(X) = \frac{12}{13}

By definition, the cosine of an acute angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. For angle

X

, the adjacent side is

XY

and the hypotenuse is

XZ

, so

\cos(X) = \frac{XY}{XZ} = \frac{12}{13}

.

Key Concept

Using the area formula, Pythagorean theorem, and trigonometric ratios to solve right triangles.

Question 165Question →

In the $xy$ -plane, a circle is represented by the equation $x^2 + y^2 - 10x + 8y - 8 = 0$ . What is the diameter of this circle?

7

14

49

98

Show answer & explanation

Answer: 14

Answer

14

To find the diameter of the circle, we first rewrite the equation in standard form,

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

, by completing the square. Grouping the terms gives

(x^2 - 10x) + (y^2 + 8y) = 8

. Adding

25

and

16

to both sides yields

(x - 5)^2 + (y + 4)^2 = 49

. Since

r^2 = 49

, the radius of the circle is

7

. The diameter is twice the radius, which is

14

.

Step-by-Step Solution

1

Group the

x

-terms and

y

-terms and move the constant to the right side of the equation.

(x^2 - 10x) + (y^2 + 8y) = 8

This prepares the equation for completing the square for both variables.

2

Complete the square for both the

x

and

y

expressions by adding

( \frac{-10}{2} )^2 = 25

and

( \frac{8}{2} )^2 = 16

to both sides of the equation.

(x^2 - 10x + 25) + (y^2 + 8y + 16) = 8 + 25 + 16

Completing the square converts the equation into the standard form of a circle's equation.

3

Rewrite the left side as squared binomials and simplify the right side.

(x - 5)^2 + (y + 4)^2 = 49

This puts the equation in the standard form

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

.

4

Identify

r^2

from the standard form equation and calculate the radius

r

.

r^2 = 49 \implies r = 7

In the standard form equation of a circle, the constant on the right side is the square of the radius.

5

Calculate the diameter by doubling the radius.

Diameter

= 2r = 2(7) = 14

The diameter of a circle is twice its radius.

Key Concept

Equations of Circles in the Coordinate Plane

Question 166Question →

A rectangular field has a diagonal path of length $40$ meters. The length of the field is $8$ meters greater than its width. What is the width, in meters, of the field?

Show answer & explanation

Answer: 24

Answer

The correct answer is 24.

Letting

w

represent the width of the rectangular field in meters, the length is

w + 8

meters. The diagonal forms the hypotenuse of a right triangle with the width and the length as legs. By the Pythagorean theorem,

w^2 + (w+8)^2 = 40^2

. Expanding this yields

2w^2 + 16w + 64 = 1600

, which simplifies to

w^2 + 8w - 768 = 0

. Factoring this equation gives

(w-24)(w+32) = 0

. Since width must be positive, the width is 24.

Step-by-Step Solution

1

Represent the dimensions of the rectangular field.

Let the width of the field be

w

meters, so the length is

w + 8

meters.

The problem states the length is 8 meters greater than the width.

2

Apply the Pythagorean theorem.

w^2 + (w + 8)^2 = 40^2

The diagonal of a rectangle forms a right triangle with the width and length as its legs.

3

Simplify the quadratic equation.

w^2 + 8w - 768 = 0

Expanding the equation gives

2w^2 + 16w - 1536 = 0

, which simplifies when divided by 2.

4

Solve for the variable

w

.

(w - 24)(w + 32) = 0

, so

w = 24

or

w = -32

.

Factoring the quadratic equation gives the possible values for the width.

5

Determine the valid physical width.

w = 24

Since the width of a field must be positive, we reject the negative solution.

Key Concept

Using the Pythagorean theorem to relate the sides of a right triangle in a geometric word problem.

Question 167Question →

A drone is hovering directly above a point on the ground. A surveyor's rangefinder, positioned $1.6$ meters above the ground, measures the angle of elevation to the drone to be $60^\circ$ . The rangefinder is located at a horizontal distance of $12\sqrt{3}$ meters from the point directly beneath the drone. What is the height, in meters, of the drone above the ground?

13.6

37.6

36.0

25.6

Show answer & explanation

Answer: $37.6$

Answer

The height of the drone above the ground is

37.6

meters.

To find the height of the drone, we first represent the situation with a right triangle. The horizontal distance from the point directly below the drone to the rangefinder is

12\sqrt{3}

meters. The angle of elevation is

60^\circ

. In this right triangle, the side opposite the

60^\circ

angle represents the vertical height of the drone above the level of the rangefinder, which we can call

h

. Using the tangent ratio, we have

\tan(60^\circ) = \frac{h}{12\sqrt{3}}

. Since

\tan(60^\circ) = \sqrt{3}

, we find

h = 12\sqrt{3} \times \sqrt{3} = 36

meters. Finally, we add the height of the rangefinder above the ground to find the total height of the drone:

36 + 1.6 = 37.6

meters.

Step-by-Step Solution

1

Identify the right triangle components from the given scenario.

A right triangle is formed where the horizontal leg (adjacent to the

60^\circ

angle) has a length of

12\sqrt{3}

meters, and the vertical leg (opposite the

60^\circ

angle) represents the height

h

of the drone above the level of the rangefinder.

This sets up the geometric model to use trigonometric ratios.

2

Use the tangent trigonometric ratio to find the vertical height

h

above the rangefinder level.

\tan(60^\circ) = \frac{h}{12\sqrt{3}} \Rightarrow \sqrt{3} = \frac{h}{12\sqrt{3}} \Rightarrow h = 12\sqrt{3} \times \sqrt{3} = 36

meters.

The tangent function relates the opposite side to the adjacent side in a right triangle.

3

Calculate the total height of the drone above the ground by adding the height of the rangefinder.

\text{Total Height} = 36 + 1.6 = 37.6

meters.

The rangefinder is positioned

1.6

meters above the ground, so the drone's height relative to the ground is the sum of its height above the rangefinder and the rangefinder's height.

Key Concept

Solving right triangles using trigonometric ratios (specifically tangent) or special 30-60-90 right triangle properties, and accounting for the height of the observer.

Question 168Question →

In the figure below, triangle $ABC$ is a right triangle with the right angle at $B$ . Point $D$ lies on side $AC$ such that segment $BD$ is perpendicular to side $AC$ . If $\cos(A) = \frac{4}{5}$ and the length of segment $AD$ is $16$ , what is the length of segment $CD$ ?

8

9

12

15

Show answer & explanation

Answer: 9

Answer

The length of segment CD is 9.

The correct answer is the option indicating that the length of segment CD is 9. By applying the definition of cosine in right triangle ADB, the hypotenuse AB is found to be 20. The Pythagorean theorem then yields the length of the altitude BD as 12. Because triangle ADB is similar to triangle BDC, the ratio of the shorter leg to the longer leg is consistent between the triangles, giving the proportion CD/BD = BD/AD. Solving this proportion results in CD = 9.

Step-by-Step Solution

1

Find the length of AB using the definition of cosine in right triangle ADB.

AB = 20

In right triangle ADB,

\cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AD}{AB}

. Substituting the given values:

\frac{4}{5} = \frac{16}{AB}

, which simplifies to

AB = 20

.

2

Find the length of BD using the Pythagorean theorem in right triangle ADB.

BD = 12

Since ADB is a right triangle,

AD^2 + BD^2 = AB^2

. Substituting the known lengths:

16^2 + BD^2 = 20^2 \implies 256 + BD^2 = 400 \implies BD^2 = 144 \implies BD = 12

.

3

Use similar triangles to set up a proportion and solve for CD.

CD = 9

Triangles ADB and BDC are similar. Comparing the ratio of the shorter leg to the longer leg in both triangles gives

\frac{CD}{BD} = \frac{BD}{AD}

. Substituting the known values:

\frac{CD}{12} = \frac{12}{16} \implies CD = 12 \times \frac{3}{4} = 9

.

Key Concept

Using trigonometry and similar right triangles to find unknown segment lengths.

Question 169Question →

In the $xy$ -plane, a circle with center at the origin passes through the point $(0, 8)$ . A line tangent to the circle at point $T$ passes through the point $P(15, 8)$ . What is the length of segment $PT$ ?

Show answer & explanation

Answer: 15

Answer

The length of segment

PT

is

15

.

The radius of the circle is

8

because the circle is centered at

(0,0)

and passes through

(0,8)

. The distance from the center

O(0,0)

to point

P(15,8)

is

OP = \sqrt{15^2 + 8^2} = 17

. The radius

OT

is perpendicular to the tangent segment

PT

at point

T

, forming a right triangle

\triangle OTP

with hypotenuse

OP

and legs

OT

and

PT

. Using the Pythagorean theorem,

PT = \sqrt{17^2 - 8^2} = 15

.

Step-by-Step Solution

1

Determine the radius of the circle

Radius

r = 8

The circle is centered at the origin

(0, 0)

and passes through

(0, 8)

, so the distance from the center to this point is the radius.

2

Calculate the distance from the origin

O(0,0)

to the point

P(15, 8)

Distance

OP = 17

Using the distance formula in the coordinate plane:

OP = \sqrt{(15-0)^2 + (8-0)^2} = \sqrt{225 + 64} = 17

.

3

Apply the Pythagorean theorem to the right triangle

\triangle OTP

Length

PT = 15

Since the tangent line

PT

is perpendicular to the radius

OT

at the point of tangency

T

,

\triangle OTP

is a right triangle with hypotenuse

OP = 17

and leg

OT = 8

. Thus,

PT = \sqrt{OP^2 - OT^2} = \sqrt{17^2 - 8^2} = 15

.

Key Concept

Tangent lines to circles and the Pythagorean theorem in the coordinate plane

Question 170Question →

In the $xy$ -plane, the graph of the equation $x^2 + y^2 - 8x + 6y - 11 = 0$ is a circle. If the center of the circle is $(h, k)$ and the radius is $r$ , what is the value of $h + k + r$ ?

Show answer & explanation

Answer: 7

Answer

7

To find

h + k + r

, we convert the general equation

x^2 + y^2 - 8x + 6y - 11 = 0

to standard form

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

by completing the square. Grouping terms gives

(x^2 - 8x) + (y^2 + 6y) = 11

. Adding 16 and 9 to both sides to complete the square yields

(x^2 - 8x + 16) + (y^2 + 6y + 9) = 11 + 16 + 9

, which simplifies to

(x - 4)^2 + (y + 3)^2 = 36

. Comparing this to the standard form gives center

(h, k) = (4, -3)

and radius

r = \sqrt{36} = 6

. The sum of these values is

4 + (-3) + 6 = 7

.

Step-by-Step Solution

1

Group the variable terms and move the constant term to the right side of the equation.

(x^2 - 8x) + (y^2 + 6y) = 11

To arrange the equation for completing the square.

2

Complete the square for both variables by adding the square of half the coefficient of the linear terms to both sides.

(x^2 - 8x + 16) + (y^2 + 6y + 9) = 11 + 16 + 9

This creates factorable perfect square trinomials on the left side while maintaining equality.

3

Factor the perfect square trinomials and add the constants on the right side.

(x - 4)^2 + (y + 3)^2 = 36

To write the equation in the standard form of a circle:

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

.

4

Determine the coordinates of the center

(h, k)

and the radius

r

from the standard form.

Center coordinates

h = 4

,

k = -3

, and radius

r = 6

Comparing

(x - 4)^2 + (y + 3)^2 = 36

to standard form reveals

h = 4

,

k = -3

, and

r = \sqrt{36} = 6

.

5

Sum the values of

h

,

k

, and

r

together.

4 + (-3) + 6 = 7

To find the final requested value.

Key Concept

Converting the general equation of a circle into standard form by completing the square to identify its center and radius.

Question 171Question →

What is the area, in square units, of the circle in the $xy$ -plane represented by the equation $x^2 + y^2 + 6x - 4y - 12 = 0$ ?

5\pi

12\pi

\pi

25\pi

Show answer & explanation

Answer: $25\pi$

Answer

The correct answer is

25\pi

.

To find the area of the circle, the equation is rewritten in the standard form

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

. Rearranging the terms gives

(x^2 + 6x) + (y^2 - 4y) = 12

. Completing the square by adding 9 and 4 to both sides yields

(x + 3)^2 + (y - 2)^2 = 12 + 9 + 4 = 25

. In this standard form, the squared radius of the circle is

r^2 = 25

. Because the area of a circle is calculated as

A = \pi r^2

, the area is

25\pi

.

Step-by-Step Solution

1

Group the

x

-terms and

y

-terms, and move the constant term to the right side of the equation.

(x^2 + 6x) + (y^2 - 4y) = 12

This prepares the equation for completing the square for both variables.

2

Complete the square for the

x

-terms by adding

(6/2)^2 = 9

to both sides, and for the

y

-terms by adding

(-4/2)^2 = 4

to both sides.

(x^2 + 6x + 9) + (y^2 - 4y + 4) = 12 + 9 + 4

, which simplifies to

(x + 3)^2 + (y - 2)^2 = 25

.

This expresses the equation in the standard form of a circle,

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

, where the center is

(h, k)

and the radius is

r

.

3

Identify the squared radius

r^2

from the standard form and calculate the area of the circle.

r^2 = 25

, so the area of the circle is

\pi r^2 = 25\pi

.

The area of a circle is calculated using the formula

A = \pi r^2

, so we directly substitute the value of

r^2

as 25.

Key Concept

Converting the general equation of a circle to standard form to find its radius and calculate its area.

Question 172Question →

In the $xy$ -plane, the graph of the equation $x^2 + y^2 - 12x + 10y + k = 0$ , where $k$ is a constant, is a circle. If this circle is tangent to the $y$ -axis, what is the value of $k$ ?

Show answer & explanation

Answer: 25

Answer

The value of

k

is 25.

Completing the square for the equation

x^2 + y^2 - 12x + 10y + k = 0

yields standard form

(x - 6)^2 + (y + 5)^2 = 61 - k

. The center of the circle is

(6, -5)

and the radius squared is

r^2 = 61 - k

. Since the circle is tangent to the

y

-axis (the line

x = 0

), its radius must be equal to the horizontal distance from the center to the

y

-axis, which is the absolute value of the

x

-coordinate of the center,

|6| = 6

. Therefore, the radius is

6

, and the radius squared is

r^2 = 36

. Setting

61 - k = 36

gives

k = 25

.

Step-by-Step Solution

1

Complete the square for

x

and

y

variables in the equation.

(x - 6)^2 + (y + 5)^2 = 61 - k

Completing the square allows the general form equation of the circle to be written in standard form, where the center and radius are easily identifiable.

2

Determine the center and the expression for the radius squared from the standard form.

Center is

(6, -5)

and

r^2 = 61 - k

In the standard circle equation

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

, the center is

(h, k_c)

and the right side represents the radius squared.

3

Relate the tangency condition to the radius of the circle.

r = 6

A circle tangent to the

y

-axis has its radius equal to the distance from its center to the line

x = 0

, which is the absolute value of the center's

x

-coordinate.

4

Calculate the value of the constant

k

.

k = 25

Since

r = 6

, then

r^2 = 36

. Setting

61 - k = 36

and solving for

k

gives

k = 25

.

Key Concept

Completing the square to find the standard form of a circle's equation and applying geometric tangency properties to determine coordinates or constants.

Question 173Question →

In right triangle $PQR$ , the measure of angle $PQR$ is $90^\circ$ . Point $S$ lies on side $PQ$ . If the length of side $QR$ is $15$ , the length of segment $SR$ is $17$ , and the length of segment $PS$ is $12$ , what is the length of side $PR$ ?

Show answer & explanation

Answer: 25

Answer

The length of side

PR

is

25

.

First, the Pythagorean theorem is applied to the smaller right triangle

SQR

to solve for the unknown leg

QS

:

QS = \sqrt{SR^2 - QR^2} = \sqrt{17^2 - 15^2} = 8

. Next, the total length of leg

PQ

is the sum of its parts,

PS

and

QS

, which gives

PQ = 12 + 8 = 20

. Finally, the Pythagorean theorem is applied to the main right triangle

PQR

to find the hypotenuse

PR

:

PR = \sqrt{PQ^2 + QR^2} = \sqrt{20^2 + 15^2} = \sqrt{400 + 225} = \sqrt{625} = 25

.

Step-by-Step Solution

1

Use the Pythagorean theorem on the right triangle

SQR

to find the length of segment

QS

.

QS = 8

Triangle

SQR

shares the right angle at

Q

with triangle

PQR

. Thus,

QS^2 + QR^2 = SR^2 \implies QS^2 + 15^2 = 17^2 \implies QS = 8

.

2

Determine the length of side

PQ

.

PQ = 20

Since

S

lies on side

PQ

, the length of

PQ

is the sum of

PS

and

QS

:

PQ = 12 + 8 = 20

.

3

Use the Pythagorean theorem on right triangle

PQR

to calculate the length of the hypotenuse

PR

.

PR = 25

PQ^2 + QR^2 = PR^2 \implies 20^2 + 15^2 = PR^2 \implies 400 + 225 = 625 \implies PR = 25

.

Key Concept

Multi-step application of the Pythagorean theorem in nested right triangles

Question 174Question →

An isosceles trapezoid has a shorter base of length $10$ , a longer base of length $22$ , and a height of $8$ . What is the perimeter of the trapezoid?

Show answer & explanation

Answer: 52

Answer

The perimeter of the trapezoid is

52

.

To find the perimeter of the isosceles trapezoid, we first determine the length of the two congruent slanted sides. By drawing altitudes from the endpoints of the shorter top base to the longer bottom base, we create two right triangles at the sides of the trapezoid. The vertical leg of each right triangle is equal to the height of the trapezoid (

8

). The horizontal leg of each right triangle is equal to half the difference between the two base lengths, which is

\frac{22 - 10}{2} = 6

. Applying the Pythagorean theorem, the length of each slanted side is

\sqrt{6^2 + 8^2} = 10

. Finally, the perimeter is the sum of all four sides:

10 + 22 + 10 + 10 = 52

.

Step-by-Step Solution

1

Calculate the horizontal base of the right triangles formed by drawing heights from the top base to the bottom base.

6

An isosceles trapezoid is symmetrical. Drawing vertical lines representing the height of

8

from the endpoints of the shorter base of length

10

down to the longer base of length

22

divides the longer base into a middle segment of length

10

and two equal end segments. The length of each end segment is

\frac{22 - 10}{2} = 6

.

2

Use the Pythagorean theorem to find the length of the slanted legs of the trapezoid.

10

Each slanted leg is the hypotenuse of a right triangle with legs of length

6

and

8

. According to the Pythagorean theorem, the hypotenuse length is

\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10

.

3

Find the perimeter of the trapezoid by adding the lengths of all four sides.

52

The perimeter is the sum of the shorter base (

10

), the longer base (

22

), and the two slanted legs (each of length

10

). Therefore, the perimeter is

10 + 22 + 10 + 10 = 52

.

Key Concept

Applying the Pythagorean theorem to find missing side lengths in composite geometric figures

Question 175Question →

A $25$ -foot ladder is leaning against a vertical wall such that the base of the ladder is $7$ feet away from the base of the wall. If the top of the ladder slides down the wall by $4$ feet, how many feet does the base of the ladder slide outward along the ground?

4

8

9

15

Show answer & explanation

Answer: $8$

Answer

The base of the ladder slides outward by

8

feet.

The correct answer is

8

feet. By applying the Pythagorean theorem, the initial height of the ladder is found to be

\sqrt{25^2 - 7^2} = 24

feet. When the ladder slides down

4

feet, its new height is

20

feet. Using the Pythagorean theorem again, the new base distance is

\sqrt{25^2 - 20^2} = 15

feet. The difference between the new base distance and the initial base distance is

15 - 7 = 8

feet.

Step-by-Step Solution

1

Find the initial height of the top of the ladder on the wall using the Pythagorean theorem.

\text{Initial Height} = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24

feet

The wall, the ground, and the ladder form a right triangle where the ladder is the hypotenuse (

25

feet) and the base distance is one leg (

7

feet).

2

Calculate the new height of the top of the ladder after it slides down.

\text{New Height} = 24 - 4 = 20

feet

The top of the ladder slides down by

4

feet from its initial height of

24

feet.

3

Calculate the new distance of the base of the ladder from the wall using the Pythagorean theorem.

\text{New Base Distance} = \sqrt{25^2 - 20^2} = \sqrt{625 - 400} = \sqrt{225} = 15

feet

The ladder length (hypotenuse) remains

25

feet, and the new height (one leg) is

20

feet.

4

Find the distance the base of the ladder slid outward by subtracting the initial base distance from the new base distance.

\text{Outward Slide} = 15 - 7 = 8

feet

The base was initially

7

feet from the wall and is now

15

feet from the wall.

Key Concept

Applying the Pythagorean theorem to relate side lengths in changing right triangle configurations.

Estimated Time:1m 30s

Question 176Question →

In the $xy$ -plane, a circle is defined by the equation $x^2 + y^2 + kx - 8y = 9$ , where $k$ is a constant. If the center of the circle lies on the line $y = 2x$ , what is the value of $k$ ?

-16

-4

-2

4

Show answer & explanation

Answer: -4

Answer

-4

The correct answer is the value that satisfies the relation between the circle's center coordinates and the given line. By completing the square on the circle's equation, we rewrite it as

(x + k/2)^2 + (y - 4)^2 = 25 + k^2/4

. This reveals the center of the circle is at the point (-k/2, 4). Substituting these coordinates into the equation of the line, which is y = 2x, gives the equation 4 = 2(-k/2). Simplifying this relation gives 4 = -k, which yields k = -4.

Step-by-Step Solution

1

Group the x and y terms of the circle equation.

(x^2 + kx) + (y^2 - 8y) = 9

Grouping terms helps in completing the square for x and y separately.

2

Complete the square for both the x and y quadratic expressions.

(x + \frac{k}{2})^2 - \frac{k^2}{4} + (y - 4)^2 - 16 = 9

, which simplifies to

(x + \frac{k}{2})^2 + (y - 4)^2 = 25 + \frac{k^2}{4}

This puts the equation into the standard circle form

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

.

3

Identify the coordinates of the center of the circle.

Center:

(h, j) = (-\frac{k}{2}, 4)

In standard form, the terms

(x - h)^2

and

(y - j)^2

correspond to a center at

(h, j)

.

4

Substitute the center coordinates into the line equation

y = 2x

to solve for

k

.

4 = 2(-\frac{k}{2}) \implies 4 = -k \implies k = -4

Since the center lies on the line, the coordinates of the center must satisfy the line's equation.

Key Concept

Completing the square to find the center of a circle in the coordinate plane and using coordinate geometry relations.

Alternative Method

Instead of completing the square fully, recall that for any circle equation of the form

x^2 + y^2 + Dx + Ey + F = 0

, the center coordinates

(h, j)

are given by

(-D/2, -E/2)

. In the given equation,

D = k

and

E = -8

. Therefore, the center is

(-\frac{k}{2}, 4)

. Substituting these coordinates directly into the line equation

y = 2x

yields

4 = 2(-\frac{k}{2})

, which simplifies to

k = -4

.

Estimated Time:1m 30s

Question 177Question →

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ . If $\sin A = \frac{5}{13}$ and the perimeter of the triangle is $60$ , what is the length of the shortest side of triangle $ABC$ ?

5

10

12

24

Show answer & explanation

Answer: 10

Answer

10

The trigonometric ratio

\sin A = \frac{5}{13}

gives the ratio of the side opposite to angle

A

to the hypotenuse. Using the Pythagorean theorem, the adjacent side corresponds to the 5-12-13 right triangle ratio. The perimeter in terms of the ratio factor

k

is

5k + 12k + 13k = 30k

. Setting

30k = 60

yields

k = 2

. The shortest side of the triangle corresponds to the smallest ratio value, which is

5k

. Multiplying

5

by the scale factor

2

yields a length of

10

.

Step-by-Step Solution

1

Use the definition of sine to set up the ratio of the side lengths.

Since

\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AB} = \frac{5}{13}

, let the side opposite to angle

A

(

BC

) be

5k

and the hypotenuse (

AB

) be

13k

, where

k

is a positive constant.

This relates the given trigonometric ratio to the side lengths of the right triangle.

2

Find the third side using the Pythagorean theorem or Pythagorean triples.

Using the Pythagorean theorem,

AC^2 + BC^2 = AB^2

, we get

AC^2 + (5k)^2 = (13k)^2 \implies AC^2 + 25k^2 = 169k^2 \implies AC^2 = 144k^2 \implies AC = 12k

. This matches the 5-12-13 Pythagorean triple.

To find the perimeter, all three side lengths of the right triangle are needed.

3

Set up an equation for the perimeter and solve for

k

.

The perimeter is the sum of all sides:

5k + 12k + 13k = 30k

. We are given that the perimeter is

60

. Thus,

30k = 60 \implies k = 2

.

Using the given perimeter helps determine the exact side lengths.

4

Determine the length of the shortest side.

The side lengths are

BC = 5(2) = 10

,

AC = 12(2) = 24

, and

AB = 13(2) = 26

. The shortest side is

BC

, which has a length of

10

.

The question asks for the length of the shortest side.

Key Concept

Using trigonometric ratios and the Pythagorean theorem to find side lengths of right triangles from perimeter.

Estimated Time:1m 30s

Question 178Question →

In the $xy$ -plane, triangle $ABC$ is a right triangle with the right angle at $B$ . The vertices of the triangle are $A(3, 11)$ , $B(15, 20)$ , and $C(27, y)$ , where $y$ is a constant. If the length of the hypotenuse $AC$ is 25 and $y < 20$ , what is the value of $y$ ?

Show answer & explanation

Answer: 4

Answer

The value of

y

is 4.

Applying the distance formula to vertices

A(3, 11)

and

B(15, 20)

yields a length of 15 for side

AB

. Since triangle

ABC

has its right angle at

B

, the hypotenuse is

AC = 25

. Using the Pythagorean theorem (

AB^2 + BC^2 = AC^2

), we find that

BC = 20

. Setting the distance between

B(15, 20)

and

C(27, y)

equal to 20 gives

(27-15)^2 + (y-20)^2 = 20^2

. This simplifies to

(y-20)^2 = 256

, which has solutions

y = 36

or

y = 4

. Given that

y < 20

, the value of

y

must be 4.

Step-by-Step Solution

1

Calculate the length of segment

AB

using the distance formula.

AB = 15

To find one of the leg lengths of the right triangle.

2

Apply the Pythagorean theorem to find the length of leg

BC

.

BC = 20

To determine the length of the other leg of the right triangle.

3

Use the distance formula for

BC

and the condition

y < 20

to solve for

y

.

y = 4

To find the vertical coordinate of vertex

C

.

Key Concept

Using the Pythagorean theorem in coordinate geometry to solve for unknown coordinates.

Geometry and Trigonometry

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