Geometry and Trigonometry Practice Questions — SAT — Page 7

Question 121Question →

In triangle $PQR$ , a line segment parallel to $QR$ intersects sides $PQ$ and $PR$ at $S$ and $T$ , respectively. If the length of segment $PS$ is $2x - 1$ , the length of segment $SQ$ is $3$ , the length of segment $PT$ is $x + 2$ , and the length of segment $TR$ is $2$ , what is the length of segment $PS$ ?

2.5

8

15

13

Show answer & explanation

Answer: 15

Answer

The length of segment

PS

is 15.

According to the Triangle Proportionality Theorem, if a line is parallel to one side of a triangle and intersects the other two sides, then it divides the two sides proportionally. Therefore, we can write the proportion as

\frac{PS}{SQ} = \frac{PT}{TR}

. Substituting the given values yields

\frac{2x - 1}{3} = \frac{x + 2}{2}

. Cross-multiplying gives

2(2x - 1) = 3(x + 2)

, which simplifies to

4x - 2 = 3x + 6

. Subtracting

3x

and adding

2

to both sides results in

x = 8

. Substituting

x = 8

back into the expression for

PS

gives

2(8) - 1 = 15

. Thus, the length of segment

PS

is 15.

Step-by-Step Solution

1

Set up a proportion using the Triangle Proportionality Theorem.

\frac{PS}{SQ} = \frac{PT}{TR}

Since segment

ST

is parallel to side

QR

, it divides the sides of triangle

PQR

proportionally.

2

Substitute the given algebraic expressions into the proportion.

\frac{2x - 1}{3} = \frac{x + 2}{2}

This establishes a solvable equation for

x

based on the geometric relationship.

3

Cross-multiply and solve for

x

.

2(2x - 1) = 3(x + 2) \implies 4x - 2 = 3x + 6 \implies x = 8

Cross-multiplication removes the denominators, allowing us to isolate the variable

x

.

4

Calculate the length of segment

PS

using the value of

x

.

PS = 2(8) - 1 = 15

The question asks for the length of segment

PS

, so we must evaluate the expression

2x - 1

at

x = 8

.

Key Concept

Triangle Proportionality Theorem and Similarity

Question 122Question →

A circular tabletop is designed to have a square glass inlay. The square glass inlay is inscribed in the circle and has an area of $64$ square inches. What is the area, in square inches, of the circular tabletop?

16\pi

32\pi

64\pi

128\pi

Show answer & explanation

Answer: $32\pi$

Answer

The area of the circular tabletop is

32\pi

square inches.

The correct answer is

32\pi

because the inscribed square has a side length of

8

inches. The diagonal of the square serves as the diameter of the circle, which is

8\sqrt{2}

inches. The radius is therefore

4\sqrt{2}

inches, and the area of the circle is

\pi (4\sqrt{2})^2 = 32\pi

square inches.

Step-by-Step Solution

1

Find the side length of the inscribed square from its area.

The side length of the square is

8

inches.

The area of a square is given by

s^2

, where

s

is the side length. Since the area is

64

, we solve

s^2 = 64

to find

s = 8

.

2

Find the length of the diagonal of the square, which represents the diameter of the circle.

The diagonal length is

8\sqrt{2}

inches.

In a square with side length

s

, the diagonal is

s\sqrt{2}

. Since the square is inscribed in the circle, the diagonal of the square is equal to the diameter of the circle.

3

Calculate the radius of the circle and then the area.

The radius is

4\sqrt{2}

inches and the area is

32\pi

square inches.

The radius

r

is half of the diameter, so

r = \frac{8\sqrt{2}}{2} = 4\sqrt{2}

. The area of the circle is given by

A = \pi r^2 = \pi (4\sqrt{2})^2 = 32\pi

.

Key Concept

Relating the area of an inscribed polygon to the circumscribed circle

Estimated Time:1m 30s

Question 123Question →

Angle $A$ has a measure of $d$ degrees, and angle $B$ has a measure of $r$ radians. The sum of the degree measure of angle $A$ and the degree equivalent of the measure of angle $B$ is $180$ . If the measure of angle $A$ , in degrees, is $3$ times the degree equivalent of the measure of angle $B$ , and $r = k\pi$ , what is the value of $k$ ?

Show answer & explanation

Answer: 0.25

Answer

The value of

k

is

0.25

(or

\frac{1}{4}

).

The correct answer is

0.25

(or

\frac{1}{4}

). To find this, we solve the system of equations representing the degree relationship:

d + D_B = 180

and

d = 3D_B

, where

D_B

is the degree equivalent of angle

B

. This gives

D_B = 45

. Converting

45^\circ

to radians by multiplying by

\frac{\pi}{180}

gives

\frac{\pi}{4}

radians, which means the coefficient

k

is

0.25

.

Step-by-Step Solution

1

Set up a system of equations using the given information.

d + D_B = 180

and

d = 3D_B

, where

d

is the degree measure of angle

A

and

D_B

is the degree equivalent of angle

B

.

To translate the verbal descriptions of the relationships between the angle measures into mathematical equations.

2

Solve the system of equations for

D_B

.

D_B = 45

Substituting

d = 3D_B

into the first equation yields

3D_B + D_B = 180

, which simplifies to

4D_B = 180

. Dividing both sides by

4

gives

D_B = 45

.

3

Convert the degree measure of angle

B

to radians.

r = \frac{\pi}{4}

radians

To convert degrees to radians, multiply the degree measure by

\frac{\pi}{180}

. This gives

45 \times \frac{\pi}{180} = \frac{\pi}{4}

.

4

Find the value of

k

from the expression

r = k\pi

.

k = 0.25

(or

\frac{1}{4}

)

Since

r = \frac{\pi}{4} = 0.25\pi

, comparing this to

r = k\pi

shows that

k = 0.25

.

Key Concept

Converting degrees to radians

Question 124Question →

In the $xy$ -plane, a circle with its center at the origin $O(0,0)$ is tangent to a line at the point $T(3.6, 4.8)$ . The line intersects the $x$ -axis at point $P$ and the $y$ -axis at point $Q$ . What is the perimeter of triangle $OPQ$ ?

14.4

24.0

30.0

35.0

Show answer & explanation

Answer: 30.0

Answer

The perimeter of triangle

OPQ

is

30.0

.

The correct answer is 30.0. By finding the distance from the origin to the point of tangency

T(3.6, 4.8)

, we find the radius is

6

. Since the tangent line is perpendicular to this radius, we can determine its equation to be

y = -0.75x + 7.5

. The intercepts are

P(10,0)

and

Q(0,7.5)

, representing the legs of right triangle

OPQ

. Applying the Pythagorean theorem, the hypotenuse is

PQ = \sqrt{10^2 + 7.5^2} = 12.5

. The sum of the sides is

10 + 7.5 + 12.5 = 30

.

Step-by-Step Solution

1

Find the radius of the circle, which is the segment

OT

from the origin

O(0,0)

to the point of tangency

T(3.6, 4.8)

.

The radius

OT = \sqrt{3.6^2 + 4.8^2} = \sqrt{12.96 + 23.04} = \sqrt{36} = 6

.

Since the line is tangent to the circle at

T

, the radius

OT

is perpendicular to the tangent line at

T

.

2

Find the equation of the tangent line. The slope of the radius

OT

is

\frac{4.8}{3.6} = \frac{4}{3}

.

The slope of the tangent line is the negative reciprocal,

-\frac{3}{4}

. Using the point-slope form with

T(3.6, 4.8)

, the equation is

y - 4.8 = -\frac{3}{4}(x - 3.6)

, which simplifies to

y = -0.75x + 7.5

.

The tangent line is perpendicular to the radius at the point of tangency.

3

Calculate the lengths of the legs of the right triangle

OPQ

by finding the

x

- and

y

-intercepts of the tangent line.

Setting

y = 0

gives the

x

-intercept

P(10, 0)

, so

OP = 10

. Setting

x = 0

gives the

y

-intercept

Q(0, 7.5)

, so

OQ = 7.5

.

The vertices

P

and

Q

lie on the axes, forming a right angle at the origin

O

, so

OP

and

OQ

are the legs of right triangle

OPQ

.

4

Use the Pythagorean theorem to find the length of the hypotenuse

PQ

, and then calculate the perimeter.

PQ = \sqrt{10^2 + 7.5^2} = \sqrt{100 + 56.25} = \sqrt{156.25} = 12.5

. The perimeter of triangle

OPQ

is

OP + OQ + PQ = 10 + 7.5 + 12.5 = 30

.

The perimeter is the sum of all three side lengths of the right triangle.

Key Concept

Right Triangles, the Pythagorean Theorem, and Tangent Lines in the Coordinate Plane

Question 125Question →

In square $ABCD$ , the side length is $12$ . Point $E$ lies on side $AB$ such that $AE = 3BE$ , and point $F$ lies on side $AD$ such that $AF = FD$ . What is the area of triangle $CEF$ ?

Show answer & explanation

Answer: 63

Answer

63

To find the area of triangle

CEF

, we subtract the areas of the three right triangles surrounding it from the total area of square

ABCD

. The area of square

ABCD

is

12^2 = 144

. Point

E

on side

AB

splits the side of length

12

into segments

AE = 9

and

BE = 3

. Point

F

on side

AD

splits the side of length

12

into equal segments

AF = 6

and

FD = 6

. The areas of the three surrounding right triangles are:

\text{Area}(\triangle AEF) = \frac{1}{2} \times 9 \times 6 = 27

,

\text{Area}(\triangle EBC) = \frac{1}{2} \times 3 \times 12 = 18

, and

\text{Area}(\triangle FDC) = \frac{1}{2} \times 6 \times 12 = 36

. Subtracting these from the total area of the square yields

\text{Area}(\triangle CEF) = 144 - (27 + 18 + 36) = 144 - 81 = 63

.

Step-by-Step Solution

1

Determine the lengths of the segments created by points

E

and

F

on the sides of the square.

Since the square has a side length of

12

, the length of side

AB

is

12

. Given that

AE = 3BE

and

AE + BE = 12

, we can write

3BE + BE = 12 \implies 4BE = 12 \implies BE = 3

. This gives

AE = 9

. Since

F

is the midpoint of

AD

(

AF = FD

), we have

AF = FD = \frac{12}{2} = 6

.

Finding these segment lengths is necessary to compute the base and height of the right triangles at the corners of the square.

2

Calculate the areas of the three right triangles surrounding triangle

CEF

.

The area of right triangle

AEF

is

\frac{1}{2} \times AE \times AF = \frac{1}{2} \times 9 \times 6 = 27

. The area of right triangle

EBC

is

\frac{1}{2} \times BE \times BC = \frac{1}{2} \times 3 \times 12 = 18

. The area of right triangle

FDC

is

\frac{1}{2} \times FD \times CD = \frac{1}{2} \times 6 \times 12 = 36

.

These three triangles occupy the entire area of the square except for the region defined by triangle

CEF

.

3

Subtract the sum of the areas of the three right triangles from the total area of square

ABCD

.

The total area of square

ABCD

is

12^2 = 144

. The area of triangle

CEF

is

144 - (27 + 18 + 36) = 144 - 81 = 63

.

This subtraction removes the corner regions, leaving only the area of the central triangle.

Key Concept

Calculating the area of an inscribed polygon by subtracting the areas of simpler surrounding geometric shapes from a larger bounding shape.

Alternative Method

Alternatively, coordinate geometry can be used. Place the vertex

D

at the origin

(0,0)

on the coordinate plane. Then the coordinates of the vertices of the square are

D(0,0)

,

C(12,0)

,

B(12,12)

, and

A(0,12)

. Point

E

lies on segment

AB

and is located at

(9,12)

. Point

F

lies on segment

AD

and is located at

(0,6)

. The area of the triangle with vertices

C(12,0)

,

E(9,12)

, and

F(0,6)

can be found using the Shoelace Formula:

\text{Area} = \frac{1}{2} |12(12 - 6) + 9(6 - 0) + 0(0 - 12)| = \frac{1}{2} |72 + 54 + 0| = \frac{1}{2} (126) = 63

.

Estimated Time:1m 30s

Question 126Question →

In the $xy$ -plane, an angle $\theta$ is in standard position. The terminal ray of the angle is rotated counterclockwise by $\frac{7\pi}{6}$ radians, and then rotated clockwise by $45^\circ$ . If the terminal ray of the resulting angle lies on the positive $y$ -axis, which of the following could be the value of $\theta$ , in degrees?

15^\circ

195^\circ

255^\circ

285^\circ

Show answer & explanation

Answer: $285^\circ$

Answer

The value of

\theta

could be

285^\circ

.

To find the value of

\theta

, we first convert the counterclockwise rotation of

\frac{7\pi}{6}

radians into degrees. Multiplying

\frac{7\pi}{6}

by

\frac{180}{\pi}

yields

210^\circ

. A counterclockwise rotation increases the angle measure, so we add

210^\circ

. A clockwise rotation decreases the angle measure, so we subtract

45^\circ

. The resulting angle is

\theta + 210^\circ - 45^\circ = \theta + 165^\circ

. Since the terminal ray of the resulting angle lies on the positive

y

-axis, it is coterminal with

90^\circ

. Setting

\theta + 165^\circ = 90^\circ + 360^\circ k

for an integer

k

gives

\theta = -75^\circ + 360^\circ k

. For

k = 1

, we get

\theta = 285^\circ

, which is the correct value.

Step-by-Step Solution

1

Convert the counterclockwise rotation from radians to degrees.

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

To work with degrees consistently, we convert the radian measure of the rotation using the conversion factor

\frac{180}{\pi}

.

2

Set up the equation for the final angle position using the directions of rotation.

Final angle =

\theta + 210^\circ - 45^\circ = \theta + 165^\circ

Counterclockwise rotations are positive (added to

\theta

) and clockwise rotations are negative (subtracted from the result).

3

Relate the final angle to the positive

y

-axis and solve for

\theta

.

\theta + 165^\circ = 90^\circ + 360^\circ k \implies \theta = -75^\circ + 360^\circ k

. For

k = 1

,

\theta = 285^\circ

.

The positive

y

-axis corresponds to

90^\circ

in standard position. Solving for

\theta

and adding multiples of

360^\circ

yields the possible measures of the starting angle.

Key Concept

Converting angle measures between radians and degrees, determining the direction of angular rotation, and finding coterminal angles.

Question 127Question →

In the $xy$ -plane, the line $y = mx$ , where $m$ is a positive constant, is tangent to the circle defined by the equation $x^2 + y^2 - 6x - 8y + 16 = 0$ . What is the value of $m$ ?

\frac{7}{24}

\frac{24}{7}

\frac{3}{4}

\frac{7}{12}

Show answer & explanation

Answer: $\frac{7}{24}$

Answer

\frac{7}{24}

The correct answer is

\frac{7}{24}

. Standardizing the circle equation gives

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

, identifying the center as

(3, 4)

and the radius as

3

. A line

mx - y = 0

is tangent to the circle if the distance from

(3, 4)

to the line is

3

. Applying the distance formula yields

\frac{|3m - 4|}{\sqrt{m^2 + 1}} = 3

, which simplifies to

m = \frac{7}{24}

after squaring and solving for

m

.

Step-by-Step Solution

1

Complete the square for the circle's equation to find the center and radius.

The given equation

x^2 + y^2 - 6x - 8y + 16 = 0

can be rewritten as

(x-3)^2 - 9 + (y-4)^2 - 16 + 16 = 0

, which simplifies to

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

. Thus, the circle has center

(3, 4)

and radius

r = \sqrt{9} = 3

.

Converting the general form equation of a circle to standard form is necessary to find the coordinates of its center and its radius.

2

Set up the equation for the distance from the center of the circle to the tangent line.

The line is given by

y = mx

, which can be written in standard form as

mx - y = 0

. The perpendicular distance from the center

(3, 4)

to this line must equal the circle's radius

3

. Using the point-to-line distance formula:

\frac{|m(3) - (4)|}{\sqrt{m^2 + (-1)^2}} = 3

.

A line is tangent to a circle if and only if the perpendicular distance from the center of the circle to the line is equal to the radius.

3

Solve the distance equation for the constant

m

.

|3m - 4| = 3\sqrt{m^2 + 1}

. Squaring both sides gives

(3m - 4)^2 = 9(m^2 + 1) \Rightarrow 9m^2 - 24m + 16 = 9m^2 + 9

. Subtracting

9m^2

from both sides gives

-24m + 16 = 9 \Rightarrow 24m = 7 \Rightarrow m = \frac{7}{24}

.

Squaring both sides eliminates the absolute value and the radical, allowing us to isolate and solve for

m

algebraically.

Key Concept

Equations of circles and the relationship between a circle and its tangent lines in the coordinate plane

Alternative Method

The problem can also be solved using geometry and right-triangle trigonometry. The distance from the origin

O(0,0)

to the center

C(3,4)

is

5

. The radius to the point of tangency

T

is

3

, forming a right triangle

OTC

with hypotenuse

OC = 5

and leg

CT = 3

. The other leg is

OT = 4

. The angle

\theta

that

OC

makes with the positive

x

-axis has

\tan\theta = \frac{4}{3}

, and the angle

\alpha

between

OC

and

OT

has

\tan\alpha = \frac{3}{4}

. The slope of the tangent line

OT

is

m = \tan(\theta - \alpha)

. Applying the tangent subtraction formula

\tan(\theta - \alpha) = \frac{\tan\theta - \tan\alpha}{1 + \tan\theta\tan\alpha}

gives

m = \frac{7}{24}

.

Estimated Time:3m 0s

Question 128Question →

In the $xy$ -plane, the graph of the equation $3x^2 + 3y^2 - 24x + 18y - c = 0$ , where $c$ is a constant, represents a circle. If the area of the circle is $100\pi$ , what is the value of $c$ ?

75

225

300

375

Show answer & explanation

Answer: 225

Answer

225

The correct answer is 225. Dividing the given equation

3x^2 + 3y^2 - 24x + 18y - c = 0

by 3 gives

x^2 + y^2 - 8x + 6y - \frac{c}{3} = 0

. Completing the square for the

x

-terms by adding

(-8/2)^2 = 16

and for the

y

-terms by adding

(6/2)^2 = 9

to both sides results in

(x - 4)^2 + (y + 3)^2 = \frac{c}{3} + 25

. The standard equation of a circle is

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

, where

r

is the radius. Since the area of the circle is

100\pi

and the area formula is

A = \pi r^2

, the radius squared

r^2

must equal 100. Setting the right-hand side of our standard form equation equal to 100 gives

\frac{c}{3} + 25 = 100

. Solving for

c

yields

\frac{c}{3} = 75

, which gives

c = 225

.

Step-by-Step Solution

1

Divide the entire equation by the coefficient of the squared terms to normalize it.

x^2 + y^2 - 8x + 6y - \frac{c}{3} = 0

The standard form of a circle's equation requires the coefficients of

x^2

and

y^2

to be 1.

2

Complete the square for both the

x

and

y

terms.

(x^2 - 8x + 16) + (y^2 + 6y + 9) = \frac{c}{3} + 16 + 9

, which simplifies to

(x - 4)^2 + (y + 3)^2 = \frac{c}{3} + 25

Completing the square converts the general equation of a circle into the standard form

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

.

3

Use the given area of the circle to determine the radius squared,

r^2

.

r^2 = 100

The area of a circle is defined by

A = \pi r^2

. Given that the area is

100\pi

, we have

\pi r^2 = 100\pi

, which means

r^2 = 100

.

4

Equate the expression for

r^2

from the standard form to the value found from the area and solve for

c

.

\frac{c}{3} + 25 = 100 \Rightarrow \frac{c}{3} = 75 \Rightarrow c = 225

From the standard form, the right-hand side is equal to

r^2

. Equating the two expressions allows us to solve for the constant

c

.

Key Concept

To find the radius or related constants of a circle from its general equation, first divide by any common coefficient of the squared terms, then complete the square to write the equation in the standard form

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

.

Question 129Question →

In a circle with center $O$ , the central angle $AOB$ has a measure of $\frac{4\pi}{5}$ radians. Angle $BOC$ is adjacent to angle $AOB$ such that they form a straight line. What is the degree measure of angle $BOC$ ?

Show answer & explanation

Answer: 36

Answer

The degree measure of angle

BOC

is 36.

Angles

AOB

and

BOC

form a straight line, which means they are supplementary and their measures add up to

180^\circ

(or

\pi

radians). Since angle

AOB

measures

\frac{4\pi}{5}

radians, we can convert this measure to degrees first by multiplying by

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

:

\frac{4\pi}{5} \times \frac{180^\circ}{\pi} = 4 \times 36^\circ = 144^\circ

. To find the degree measure of the supplementary angle

BOC

, subtract

144^\circ

from

180^\circ

, giving

180^\circ - 144^\circ = 36^\circ

. Alternatively, the calculation can be performed in radians first:

\pi - \frac{4\pi}{5} = \frac{\pi}{5}

radians, and then converting

\frac{\pi}{5}

radians to degrees:

\frac{\pi}{5} \times \frac{180^\circ}{\pi} = 36^\circ

.

Step-by-Step Solution

1

Identify the relationship between the two adjacent angles.

The sum of the measures of angles

AOB

and

BOC

is

\pi

radians or

180^\circ

.

Since the adjacent angles

AOB

and

BOC

form a straight line, they are supplementary.

2

Calculate the measure of angle

BOC

in radians.

Angle

BOC

measures

\frac{\pi}{5}

radians.

Subtract the measure of angle

AOB

from the straight line measure:

\pi - \frac{4\pi}{5} = \frac{\pi}{5}

radians.

3

Convert the radian measure of angle

BOC

to degrees.

The degree measure is 36.

Multiply the radian measure by the conversion factor

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

:

\frac{\pi}{5} \times \frac{180}{\pi} = 36^\circ

.

Key Concept

Converting angles from radians to degrees and using the properties of supplementary angles.

Question 130Question →

For triangle $ABC$ , points $D$ and $E$ are on sides $AB$ and $AC$ , respectively, such that segment $DE$ is parallel to segment $BC$ . If the length of segment $AD$ is $x + 5$ , the length of segment $DB$ is $6$ , the length of segment $AE$ is $x - 2$ , and the length of segment $EC$ is $3$ . What is the length of segment $AE$ ?

2

5

7

9

Show answer & explanation

Answer: 7

Answer

7

According to the Triangle Proportionality Theorem, since segment

DE

is parallel to segment

BC

, the sides of triangle

ABC

are divided proportionally:

\frac{AD}{DB} = \frac{AE}{EC}

. Substituting the given expressions, we get

\frac{x+5}{6} = \frac{x-2}{3}

. Multiplying both sides of the equation by 6 gives

x+5 = 2(x-2)

. Distributing the 2 to both terms inside the parentheses yields

x+5 = 2x-4

. Subtracting

x

from both sides gives

5 = x-4

, and adding 4 to both sides gives

x = 9

. The question asks for the length of segment

AE

, which is defined as

x-2

. Substituting

x = 9

into this expression gives

9 - 2 = 7

.

Step-by-Step Solution

1

Identify the relationship between the triangles and set up the proportion.

Since

DE \parallel BC

, triangle

ADE

is similar to triangle

ABC

(

\triangle ADE \sim \triangle ABC

). By the Triangle Proportionality Theorem, this gives the proportion

\frac{AD}{DB} = \frac{AE}{EC}

.

A line parallel to one side of a triangle divides the other two sides proportionally.

2

Substitute the given side lengths into the proportion.

\frac{x+5}{6} = \frac{x-2}{3}

Using the algebraic expressions provided in the problem statement.

3

Solve the equation for

x

.

Multiplying both sides by 6 gives

x+5 = 2(x-2)

. Distributing the 2 gives

x+5 = 2x-4

. Subtracting

x

and adding 4 to both sides yields

x = 9

.

To determine the value of the variable

x

.

4

Calculate the length of segment

AE

.

AE = x - 2 = 9 - 2 = 7

.

The question asks for the length of segment

AE

, so we substitute

x = 9

back into the expression for

AE

.

Key Concept

Using the Triangle Proportionality Theorem and properties of similar triangles to solve for unknown side lengths using algebraic expressions.

Estimated Time:1m 30s

Question 131Question →

In a circle, chords $AB$ and $CD$ intersect perpendicularly at point $E$ . If $AE = 35$ , $EB = 5$ , and $CE = 5$ , what is the length of the diameter of the circle?

Show answer & explanation

Answer: 50

Answer

50

The correct answer is 50. By applying the intersecting chords theorem, the segment

ED

is found to be 35. Since the chords are perpendicular and intersect at point

E

, we can find the distance from the center of the circle to each chord by analyzing the distances from the intersection point to the midpoints of the chords. The midpoints of both chords are 20 units from their endpoints. The distance from

E

to the midpoint of

AB

is

35 - 20 = 15

. This distance is equal to the perpendicular distance from the center of the circle to the other chord,

CD

. Using the Pythagorean theorem with a chord half-length of 20 and a distance from the center of 15, the radius of the circle is

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

. Therefore, the diameter of the circle is

2 \times 25 = 50

.

Step-by-Step Solution

1

Find the length of segment

ED

using the intersecting chords theorem.

ED = 35

For any two intersecting chords

AB

and

CD

intersecting at point

E

, the product of the segments of one chord equals the product of the segments of the other:

AE \cdot EB = CE \cdot ED

. Substituting the given values:

35 \cdot 5 = 5 \cdot ED

, which simplifies to

ED = 35

.

2

Calculate the total lengths of chords

AB

and

CD

and determine their midpoints.

Chord lengths

AB = 40

and

CD = 40

. Midpoint distances

MB = 20

and

ND = 20

.

The total length of chord

AB

is

AE + EB = 35 + 5 = 40

. The perpendicular line from the center

O

to

AB

bisects the chord at midpoint

M

, so

MB = 40 / 2 = 20

. Similarly, the total length of chord

CD

is

CE + ED = 5 + 35 = 40

, and its midpoint

N

bisects it, so

ND = 20

.

3

Find the perpendicular distances from the center

O

to the chords

AB

and

CD

.

OM = 15

and

ON = 15

The distance from the intersection point

E

to the midpoint

M

along chord

AB

is

AE - AM = 35 - 20 = 15

. Because the chords are perpendicular, the perpendicular distance from the center

O

to chord

AB

is equal to the distance

EN

along the other chord, so

OM = EN = 15

. Similarly,

ON = EM = 15

.

4

Calculate the radius of the circle using the Pythagorean theorem.

Radius

R = 25

In the right triangle

OMB

, the hypotenuse is the radius

R = OB

, and the legs are the perpendicular distance

OM = 15

and half the chord length

MB = 20

. By the Pythagorean theorem,

R^2 = OM^2 + MB^2 = 15^2 + 20^2 = 225 + 400 = 625

. Taking the square root gives

R = 25

.

5

Calculate the diameter of the circle.

Diameter =

50

The diameter of a circle is twice its radius:

2R = 2 \cdot 25 = 50

.

Key Concept

Using perpendicular chords, the intersecting chords theorem, and the Pythagorean theorem to determine the radius and diameter of a circle.

Question 132Question →

A circular dial on a vintage radio is rotated by $\frac{5\pi}{8}$ radians to tune to a specific station. If the dial is then rotated by an additional $45^\circ$ in the same direction, what is the total angle of rotation, in radians, of the dial?

\frac{3\pi}{4}

\frac{7\pi}{8}

\frac{9\pi}{8}

\frac{\pi}{2}

Show answer & explanation

Answer: $\frac{7\pi}{8}$

Answer

The correct answer is

\frac{7\pi}{8}

radians.

To find the total angle of rotation in radians, the rotation of

45^\circ

must first be converted to radians by multiplying by

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

, which yields

\frac{\pi}{4}

radians. Expressed with a common denominator of

8

, this is equivalent to

\frac{2\pi}{8}

radians. Adding this to the initial rotation of

\frac{5\pi}{8}

radians gives a total rotation of

\frac{5\pi}{8} + \frac{2\pi}{8} = \frac{7\pi}{8}

radians.

Step-by-Step Solution

1

Convert the additional rotation angle of

45^\circ

into radians.

Since

180^\circ = \pi

radians, the conversion is

45^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{4}

radians.

To find the total rotation angle in radians, both individual angles must be in the same unit of measure.

2

Add the first rotation angle of

\frac{5\pi}{8}

radians to the converted second rotation angle of

\frac{\pi}{4}

radians.

First, express

\frac{\pi}{4}

with a common denominator of

8

:

\frac{\pi}{4} = \frac{2\pi}{8}

. Then, add the two fractions:

\frac{5\pi}{8} + \frac{2\pi}{8} = \frac{7\pi}{8}

radians.

The total angle of rotation is the sum of the two sequential rotations in the same direction.

Key Concept

To find the sum of angles given in different units, convert the angle measured in degrees to radians using the conversion factor

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

, and then find the sum of the two radian values using a common denominator.

Estimated Time:1m 0s

Question 133Question →

A landscape architect is designing a courtyard in the shape of a right trapezoid. The parallel sides of the courtyard have lengths of $24$ yards and $40$ yards. The side perpendicular to the parallel sides has a length of $15$ yards. A straight path is built from the midpoint of the longer parallel side to the vertex of the shorter parallel side that is adjacent to the perpendicular side, dividing the courtyard into two regions. What is the area, in square yards, of the smaller region?

Show answer & explanation

Answer: 150

Answer

The area of the smaller region is 150 square yards.

The straight path divides the right trapezoid into two regions: a right triangle and a quadrilateral. The right triangle has a vertical leg of 15 yards (the height of the trapezoid) and a horizontal leg of 20 yards (half of the longer parallel side of 40 yards). The area of this right triangle is 0.5 * 20 * 15 = 150 square yards. The total area of the trapezoid is 0.5 * (24 + 40) * 15 = 480 square yards, making the area of the quadrilateral region 480 - 150 = 330 square yards. Comparing the two regions, the smaller region has an area of 150 square yards.

Step-by-Step Solution

1

Find the length of the segment from the perpendicular corner to the midpoint of the longer parallel side.

20 yards

The midpoint divides the 40-yard side into two equal parts of 20 yards each.

2

Determine the shape and dimensions of the region containing the perpendicular side.

A right triangle with legs of 15 yards and 20 yards.

Since the path goes from the midpoint of the base to the opposite vertex of the perpendicular height, it forms a right triangle with the height and half of the longer base.

3

Calculate the area of this right triangle.

150 square yards

Using the area formula for a triangle, Area = 0.5 * base * height = 0.5 * 20 * 15 = 150.

4

Calculate the total area of the trapezoid and the area of the remaining region to confirm which is smaller.

Total area is 480 square yards; the other region's area is 330 square yards.

The total area is 0.5 * (24 + 40) * 15 = 480. The other region has an area of 480 - 150 = 330. Comparing 150 and 330, 150 is the smaller area.

Key Concept

Area of composite shapes and trapezoids

Question 134Question →

A circular fountain with a diameter of $8$ feet is positioned in the center of a square lawn. The lawn has a side length of $20$ feet. A concrete walkway of width $2$ feet is built directly around the fountain. What is the area, in square feet, of the remaining grass region of the lawn?

400 - 36\pi

400 - 16\pi

400 - 24\pi

400 - 100\pi

Show answer & explanation

Answer: $400 - 36\pi$

Answer

400 - 36\pi

The area of the square lawn is

20^2 = 400

square feet. The fountain has a radius of

4

feet (half of the

8

-foot diameter), and the walkway adds another

2

feet to the radius, making the total radius of the combined circular area

6

feet. The area of this circular region is

\pi \times 6^2 = 36\pi

square feet. Subtracting this from the total area of the square lawn gives the remaining grass area of

400 - 36\pi

square feet.

Step-by-Step Solution

1

Calculate the total area of the square lawn.

Area of the lawn is

20 \times 20 = 400

square feet.

The area of a square is calculated by squaring its side length.

2

Find the combined radius of the fountain and the walkway.

The radius of the fountain is

8 \div 2 = 4

feet. Adding the

2

-foot width of the walkway gives a combined radius of

4 + 2 = 6

feet.

The radius is half of the diameter. The walkway surrounds the fountain, so its width must be added to the fountain's radius to find the outer boundary's radius.

3

Calculate the combined area of the fountain and the walkway.

Area of the combined circular region is

\pi \times 6^2 = 36\pi

square feet.

The area of a circle is given by

\pi r^2

, where

r

is the radius.

4

Subtract the combined circular area from the total area of the lawn to find the remaining grass area.

The remaining area is

400 - 36\pi

square feet.

The remaining grass area is the total square area minus the area of the inner circular region that contains the fountain and the walkway.

Key Concept

Area of composite shapes involving squares and circles, and scaling properties.

Estimated Time:1m 30s

Question 135Question →

In the $xy$ -plane, the graph of the equation $x^2 + y^2 - 12x + 8y - k = 0$ , where $k$ is a positive constant, is a circle. If the line $y = 6$ is tangent to the circle, what is the value of $k$ ?

Show answer & explanation

Answer: 48

Answer

48

To find the value of

k

, we convert the given circle equation into its standard form,

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

. Grouping the terms gives

(x^2 - 12x) + (y^2 + 8y) = k

. Completing the square for

x

and

y

gives

(x - 6)^2 - 36 + (y + 4)^2 - 16 = k

, which simplifies to

(x - 6)^2 + (y + 4)^2 = k + 52

. This tells us that the center of the circle is

(6, -4)

and the radius squared is

r^2 = k + 52

. A horizontal line

y = 6

is tangent to the circle, meaning the perpendicular distance from the center

(6, -4)

to the line

y = 6

is equal to the radius. This distance is

|6 - (-4)| = 10

. Therefore, the radius is

10

, which means

r^2 = 100

. Equating the two expressions for the radius squared gives

k + 52 = 100

. Solving this equation yields

k = 48

.

Step-by-Step Solution

1

Group the variables and complete the square for the

x

and

y

terms.

(x - 6)^2 + (y + 4)^2 = k + 52

Completing the square converts the equation from general form to standard form, which reveals the center and radius.

2

Identify the center of the circle and the algebraic representation of the radius.

Center is

(6, -4)

and radius

r = \sqrt{k + 52}

.

The standard equation of a circle is

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

, where

(h, j)

is the center and

r

is the radius.

3

Find the radius of the circle using the given tangent line.

Radius

r = 10

The distance from the center's

y

-coordinate,

-4

, to the horizontal tangent line

y = 6

is

|6 - (-4)| = 10

, which represents the radius of the circle.

4

Equate the radius squared value to the algebraic expression for the radius squared and solve for

k

.

k = 48

Since the radius is

10

, the radius squared is

100

. Setting

k + 52 = 100

and solving for

k

yields

48

.

Key Concept

Converting a circle's equation from general to standard form by completing the square, and using the distance from the center to a tangent line to find the radius.

Question 136Question →

The equation $x^2 + y^2 + 10x - 6y - 15 = 0$ represents a circle in the coordinate plane. If this circle is translated $2$ units to the right and $4$ units down, which of the following equations represents the translated circle?

(x - 3)^2 + (y - 1)^2 = 49

(x + 3)^2 + (y + 1)^2 = 49

(x + 3)^2 + (y + 1)^2 = 7

(x + 7)^2 + (y - 7)^2 = 49

Show answer & explanation

Answer: $(x + 3)^2 + (y + 1)^2 = 49$

Answer

The equation of the translated circle is

(x + 3)^2 + (y + 1)^2 = 49

.

The correct answer shows

(x + 3)^2 + (y + 1)^2 = 49

. Completing the square on the original equation reveals the original center is

(-5, 3)

and

r^2 = 49

. Translating the center

2

units right and

4

units down shifts the

x

-coordinate from

-5

to

-3

and the

y

-coordinate from

3

to

-1

. The equation for a circle with center

(-3, -1)

and radius squared of

49

is

(x + 3)^2 + (y + 1)^2 = 49

.

Step-by-Step Solution

1

Group the

x

-terms and

y

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

(x^2 + 10x) + (y^2 - 6y) = 15

This groups terms by variable to prepare for completing the square.

2

Complete the square for the

x

-terms by adding

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

and for the

y

-terms by adding

(\frac{-6}{2})^2 = 9

to both sides of the equation.

(x^2 + 10x + 25) + (y^2 - 6y + 9) = 15 + 25 + 9

, which simplifies to

(x + 5)^2 + (y - 3)^2 = 49

.

This converts the circle's equation into standard form

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

.

3

Identify the center and the radius squared of the original circle from the standard form equation.

The center is

(h, k) = (-5, 3)

and the radius squared is

r^2 = 49

.

The standard form reveals the center coordinates as the opposite signs of the constants inside the parentheses, and the right side is

r^2

.

4

Calculate the center of the translated circle by shifting the original center

(-5, 3)

by

2

units to the right and

4

units down.

The new center coordinates are

(-5 + 2, 3 - 4) = (-3, -1)

.

Translating a point right increases its

x

-coordinate, and translating it down decreases its

y

-coordinate.

5

Write the standard form equation for the new circle with center

(-3, -1)

and the unchanged radius squared

r^2 = 49

.

(x - (-3))^2 + (y - (-1))^2 = 49

, which simplifies to

(x + 3)^2 + (y + 1)^2 = 49

.

Substituting

h = -3

and

k = -1

into the standard circle equation form

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

yields the final equation.

Key Concept

Completing the square to find standard circle equations and translating circle centers in the coordinate plane.

Question 137Question →

In the $xy$ -plane, the equation $2x^2 + 2y^2 - 12x + 16y - 22 = 0$ represents a circle. What is the diameter of this circle?

Show answer & explanation

Answer: 12

Answer

The diameter of the circle is 12.

Dividing the given equation

2x^2 + 2y^2 - 12x + 16y - 22 = 0

by 2 gives

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

. Grouping the

x

and

y

terms and moving the constant to the right side gives

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

. To complete the square, add

(\frac{-6}{2})^2 = 9

and

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

to both sides, yielding

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

, which simplifies to

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

. Since the standard equation of a circle is

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

, the radius squared

r^2

is 36, which means the radius

r

is 6. The diameter is twice the radius, so

2 \times 6 = 12

.

Step-by-Step Solution

1

Divide the entire equation by 2.

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

To simplify the coefficients of

x^2

and

y^2

to 1, which is necessary before completing the square.

2

Group the variables and move the constant term.

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

To isolate the quadratic and linear terms for both

x

and

y

on one side of the equation.

3

Complete the square for both variables by adding the appropriate values to both sides.

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

Adding 9 (which is

(\frac{-6}{2})^2

) and 16 (which is

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

) to both sides allows us to rewrite the trinomials as perfect squares:

(x-3)^2

and

(y+4)^2

. The right side becomes

11 + 9 + 16 = 36

.

4

Find the radius and calculate the diameter.

Diameter = 12

Comparing the equation to the standard form

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

shows that

r^2 = 36

, so the radius

r

is 6. The diameter is

2r = 2(6) = 12

.

Key Concept

Converting a circle's equation from general form to standard form by completing the square to identify its geometric properties.

Question 138Question →

In triangle $ABC$ , point $D$ lies on side $BC$ . Line segment $AD$ is drawn such that $AB = AD$ and $AD = CD$ . If the measure of angle $BAC$ is $75^\circ$ , what is the measure, in degrees, of angle $B$ ?

Show answer & explanation

Answer: 70

Answer

The measure of angle

B

is

70^\circ

.

By representing the angles using the properties of the isosceles triangles

ABD

and

ADC

, we set up a system of equations where

\angle B = x

and

\angle C = y

such that

x = 2y

. Using the angle sum theorem and angle addition, we find that

(180^\circ - 2x) + y = 75^\circ

. Substituting

x = 2y

yields

y = 35^\circ

, and thus

\angle B = x = 70^\circ

.

Step-by-Step Solution

1

Set up base angles for the isosceles triangle

ABD

.

Let

\angle B = \angle ADB = x

.

Because

AB = AD

, triangle

ABD

is an isosceles triangle, making its base angles equal.

2

Express the measure of angle

ADC

in terms of

x

.

\angle ADC = 180^\circ - x

.

Angles

\angle ADB

and

\angle ADC

form a linear pair along the line segment

BC

.

3

Establish the relationship between

x

and

y

using triangle

ADC

.

x = 2y

, where

y = \angle DAC = \angle C

.

Since

AD = CD

, triangle

ADC

is isosceles. The sum of angles in

\triangle ADC

is

(180^\circ - x) + y + y = 180^\circ

, which simplifies to

x = 2y

.

4

Write the equation for the total measure of angle

BAC

.

(180^\circ - 2x) + y = 75^\circ

.

The angle addition postulate states that

\angle BAC = \angle BAD + \angle DAC

. Since

\angle BAD = 180^\circ - 2x

(from the sum of angles in

\triangle ABD

) and

\angle DAC = y

, this equals

75^\circ

.

5

Solve the system of equations for

y

.

y = 35^\circ

.

Substituting

x = 2y

into

(180^\circ - 2x) + y = 75^\circ

yields

180^\circ - 3y = 75^\circ

, which gives

3y = 105^\circ

, so

y = 35^\circ

.

6

Find the measure of angle

B

.

\angle B = 70^\circ

.

Since

\angle B = x

and

x = 2y

, we have

x = 2(35^\circ) = 70^\circ

.

Key Concept

Using properties of isosceles triangles and angle sum theorems to solve for angle measures.

Question 139Question →

An L-shaped region is created by removing a smaller square from the corner of a larger square. The perimeter of the L-shaped region is $48$ inches, and the area of the smaller square that was removed is $16$ square inches. What is the area, in square inches, of the L-shaped region?

Show answer & explanation

Answer: 128

Answer

128

The area of the L-shaped region is the difference between the area of the original larger square and the area of the removed corner square. The perimeter of an L-shaped region formed by removing a corner square is identical to the perimeter of the original square, which is

48

inches. This means the side length of the larger square is

12

inches, and its area is

144

square units. Subtracting the area of the removed square (

16

square units) from the area of the larger square gives

128

square units.

Step-by-Step Solution

1

Calculate the side length of the smaller square from its area.

The side length of the smaller square is

4

inches.

The area of a square is the square of its side length (

A = s^2

). Since the area is

16

, we have

s = \sqrt{16} = 4

.

2

Find the side length of the larger square using the perimeter of the L-shaped region.

The side length of the larger square is

12

inches.

When a corner square is removed from a larger square, the perimeter remains unchanged because the two cut-out edges going inward have the same lengths as the two outer edges that were removed. Thus, the perimeter of the L-shaped region is equal to

4S

. With a perimeter of

48

, the side length

S

is

48 \div 4 = 12

.

3

Compute the area of the L-shaped region.

The area of the L-shaped region is

128

square inches.

The area of the L-shaped region is the area of the larger square minus the area of the removed smaller square:

12^2 - 16 = 144 - 16 = 128

.

Key Concept

The area of a composite shape can be calculated by subtracting the area of a removed sub-region from the area of the outer boundary. The perimeter of a rectangle or square remains unchanged when a corner square is removed.

Question 140Question →

In a circle with center $O$ , the radius is $6$ . Points $A$ and $B$ lie on the circle such that the area of the sector $AOB$ is $12$ . What is the length of the minor arc $AB$ ?

Show answer & explanation

Answer: 4

Answer

The length of the minor arc

AB

is 4.

The area of a sector with central angle

\theta

in radians is

A = \frac{1}{2}r^2\theta

. Setting

A = 12

and

r = 6

, we get

12 = \frac{1}{2}(6)^2\theta

, which simplifies to

12 = 18\theta

, and thus

\theta = \frac{2}{3}

radians. The length of the arc is

s = r\theta = 6 \left(\frac{2}{3}\right) = 4

.

Step-by-Step Solution

1

Find the central angle

\theta

in radians using the sector area formula.

\theta = \frac{2}{3}

The area of a sector is given by

A = \frac{1}{2}r^2\theta

, so substituting

A = 12

and

r = 6

gives

12 = 18\theta

, which yields

\theta = \frac{2}{3}

.

2

Calculate the arc length

s

using the formula

s = r\theta

.

s = 4

Substituting

r = 6

and

\theta = \frac{2}{3}

into the arc length formula gives

s = 6 \left(\frac{2}{3}\right) = 4

.

Key Concept

Calculating arc length from sector area and radius using radian measures

Geometry and Trigonometry

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