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Question 581Question →

A sector of a circle with center $O$ has an area of $54\pi$ . The perimeter of the sector is $36 + 6\pi$ . If the radius of the circle is an integer, what is the radius of the circle?

Show answer & explanation

Answer: 18

Answer

18

The area of a sector is given by

A = \frac{1}{2}rs

, where

r

is the radius and

s

is the arc length. Given

A = 54\pi

, we have

\frac{1}{2}rs = 54\pi

, which simplifies to

rs = 108\pi

, or

s = \frac{108\pi}{r}

. The perimeter of a sector is

P = 2r + s

. Given

P = 36 + 6\pi

, we can substitute

s

to get

2r + \frac{108\pi}{r} = 36 + 6\pi

. Multiplying the entire equation by

r

and rearranging terms yields the quadratic equation

2r^2 - (36 + 6\pi)r + 108\pi = 0

. Factoring by grouping gives

(2r - 6\pi)(r - 18) = 0

. This yields two potential solutions:

r = 3\pi

and

r = 18

. Since the radius is specified to be an integer, the correct answer is

18

.

Step-by-Step Solution

1

Relate sector area and perimeter to radius and arc length

Area =

\frac{1}{2}rs = 54\pi

and Perimeter =

2r + s = 36 + 6\pi

, where

r

is the radius and

s

is the arc length of the sector.

The area of a sector with radius

r

and arc length

s

is given by

\frac{1}{2}rs

, and its perimeter consists of the two radii plus the arc length.

2

Express arc length in terms of radius

s = \frac{108\pi}{r}

Isolating

s

from the area equation allows for substitution into the perimeter equation.

3

Substitute and form a quadratic equation

2r + \frac{108\pi}{r} = 36 + 6\pi \implies 2r^2 - (36 + 6\pi)r + 108\pi = 0

Multiplying both sides of the equation by

r

eliminates the fraction and forms a standard quadratic equation.

4

Factor the quadratic equation

(2r - 6\pi)(r - 18) = 0

Grouping the terms as

(2r^2 - 36r) - (6\pi r - 108\pi) = 0

allows us to factor out

2r(r - 18) - 6\pi(r - 18) = 0

.

5

Identify the integer radius

r = 18

The two solutions to the equation are

r = 3\pi

and

r = 18

. Since the problem specifies that the radius is an integer, we select

18

.

Key Concept

Calculating sector area and perimeter using relationships between radius, arc length, and angle measures.

Question 582Question →

In the $xy$ -plane, the graph of the equation $x^2 + y^2 - 14x - 12y + q = 0$ is a circle that is tangent to the $y$ -axis, where $q$ is a constant. What is the value of $q$ ?

Show answer & explanation

Answer: 36

Answer

36

To find the value of

q

, we convert the general form of the circle's equation into standard form

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

by completing the square. Grouping the variables gives

(x^2 - 14x) + (y^2 - 12y) = -q

. Adding

(14/2)^2 = 49

and

(12/2)^2 = 36

to both sides yields

(x - 7)^2 + (y - 6)^2 = 85 - q

. Thus, the center of the circle is

(7, 6)

and the radius squared is

r^2 = 85 - q

. Since the circle is tangent to the

y

-axis, its radius must equal the horizontal distance from its center to the

y

-axis, which is

7

. Therefore, the radius squared is

7^2 = 49

. Setting

85 - q = 49

and solving for

q

gives

q = 36

.

Step-by-Step Solution

1

Rearrange and group the terms of the equation to prepare for completing the square.

(x^2 - 14x) + (y^2 - 12y) = -q

Grouping terms allows us to complete the square for the

x

and

y

variables independently.

2

Complete the square for both the

x

and

y

expressions by adding the square of half the coefficient of the linear term to both sides.

(x^2 - 14x + 49) + (y^2 - 12y + 36) = -q + 49 + 36

Adding

(14/2)^2 = 49

and

(12/2)^2 = 36

transforms the trinomials into perfect squares.

3

Factor the perfect square trinomials and simplify the right side of the equation.

(x - 7)^2 + (y - 6)^2 = 85 - q

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

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

, identifying the center as

(7, 6)

and the radius squared as

r^2 = 85 - q

.

4

Determine the radius of the circle using the condition of tangency to the

y

-axis.

r = 7

A circle tangent to the

y

-axis has a radius equal to the absolute value of the x-coordinate of its center, which is

|7| = 7

.

5

Solve for the constant

q

by equating the two expressions for the radius squared.

85 - q = 49 \implies q = 36

Since

r = 7

, we have

r^2 = 49

. Setting

85 - q = 49

yields

q = 36

.

Key Concept

Completing the square to find the standard equation of a circle and applying coordinate geometry tangency conditions.

Question 583Question →

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 584Question →

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 585Question →

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 586Question →

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 587Question →

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 588Question →

In a hydroponic farming system, a nutrient solution is created by mixing a liquid fertilizer concentrate with water in a ratio of $3$ to $50$ by volume. To fill a reservoir, a technician needs to prepare a total of $318$ liters of this nutrient solution. How many liters of the liquid fertilizer concentrate are required to prepare the solution?

Show answer & explanation

Answer: 18

Answer

18

To find the amount of liquid fertilizer concentrate needed, we first express the ratio in terms of the total mixture. The ratio of concentrate to water is

3

to

50

, meaning there are

3

parts of concentrate for every

50

parts of water, making a total of

3 + 50 = 53

parts. Since the total volume of the solution is

318

liters, each part represents

\frac{318}{53} = 6

liters. The amount of concentrate required is

3

parts, which corresponds to

3 \times 6 = 18

liters.

Step-by-Step Solution

1

Calculate the total parts represented in the ratio of liquid fertilizer concentrate to water.

The total number of parts is

3 + 50 = 53

.

Since the ratio of concentrate to water is

3

to

50

, the entire nutrient solution is divided into

53

equal parts.

2

Find the volume of solution that corresponds to one part.

One part is equal to

\frac{318}{53} = 6

liters.

Dividing the total volume of the nutrient solution (

318

liters) by the total number of parts (

53

) gives the volume of a single part.

3

Calculate the total volume of liquid fertilizer concentrate needed.

The volume of concentrate required is

3 \times 6 = 18

liters.

Since the concentrate accounts for

3

parts of the total mixture, multiplying the volume of one part by

3

gives the total amount of concentrate needed.

Key Concept

Solving part-to-whole ratio problems by determining the value of a single unit or part from a given total quantity.

Alternative Method

Let the volume of the fertilizer concentrate be

3x

and the volume of water be

50x

. The total volume of the nutrient solution is

3x + 50x = 53x

. Given that the total volume is

318

liters, we can set up the equation

53x = 318

. Solving for

x

gives

x = 6

. The volume of the concentrate is

3x = 3(6) = 18

liters.

Estimated Time:1m 15s

Question 589Question →

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 590Question →

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 591Question →

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 592Question →

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 593Question →

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

Question 594Question →

In the $xy$ -plane, a circle with radius $r$ , where $r > 1$ , has its center in the first quadrant. The circle is tangent to the line $x = 1$ and tangent to the line $y = 2$ . If the center of the circle lies on the line with equation $y = \frac{4}{3}x$ , what is the value of $r$ ?

Show answer & explanation

Answer: 2

Answer

The radius

r

of the circle is 2.

Since the circle has radius

r

and is tangent to the lines

x = 1

and

y = 2

, its center

(h, k)

lies at a distance of

r

from both lines. This means

h = 1 \pm r

and

k = 2 \pm r

. Because the center is in the first quadrant,

h > 0

and

k > 0

. Given

r > 1

, the choice

h = 1 - r

would make

h

negative, so we must have

h = r + 1

. If

k = 2 - r

, the center is

(r + 1, 2 - r)

, and substituting this into the line

y = \frac{4}{3}x

gives

2 - r = \frac{4}{3}(r + 1) \implies 6 - 3r = 4r + 4 \implies 7r = 2 \implies r = \frac{2}{7}

, which contradicts the condition that

r > 1

. Therefore, we must have

k = r + 2

. Substituting the center

(r + 1, r + 2)

into

y = \frac{4}{3}x

yields

r + 2 = \frac{4}{3}(r + 1) \implies 3(r + 2) = 4(r + 1) \implies 3r + 6 = 4r + 4 \implies r = 2

.

Step-by-Step Solution

1

Set up equations for the center coordinates

(h, k)

in terms of the radius

r

.

|h - 1| = r

and

|k - 2| = r

The distance from the center of a circle to any of its tangent lines is equal to the radius

r

.

2

Determine the correct sign for the absolute value expression of the

x

-coordinate.

h = r + 1

Since the center lies in the first quadrant,

h

must be positive. If

h = 1 - r

, then

r > 1

would imply

h < 0

, which is a contradiction.

3

Determine the correct sign for the absolute value expression of the

y

-coordinate by evaluating both possibilities on the line

y = \frac{4}{3}x

.

k = r + 2

If

k = 2 - r

, then substituting into the line equation gives

r = \frac{2}{7}

, which contradicts the condition that

r > 1

. Thus,

k

must equal

r + 2

.

4

Substitute

(r + 1, r + 2)

into the line equation

y = \frac{4}{3}x

and solve for

r

.

r = 2

Substituting gives

r + 2 = \frac{4}{3}(r + 1) \implies 3(r + 2) = 4(r + 1) \implies 3r + 6 = 4r + 4 \implies r = 2

.

Key Concept

Equations of Circles in the Coordinate Plane

Question 595Question →

Triangle $ABC$ is similar to triangle $DEF$ , where the ratio of the length of side $AB$ to the length of side $DE$ is $3$ to $5$ . If the area of triangle $ABC$ is $18$ , what is the area of triangle $DEF$ ?

Show answer & explanation

Answer: 50

Answer

The area of triangle

DEF

is

50

.

Since triangle

ABC

is similar to triangle

DEF

, the ratio of their areas is the square of the ratio of their corresponding side lengths. The ratio of side

AB

to side

DE

is

3/5

, so the ratio of the area of triangle

ABC

to the area of triangle

DEF

is

(3/5)^2 = 9/25

. Given that the area of triangle

ABC

is

18

, we can set up the proportion

18 / x = 9 / 25

, where

x

represents the area of triangle

DEF

. Solving for

x

gives

x = 18 \times (25/9) = 2 \times 25 = 50

.

Step-by-Step Solution

1

Determine the ratio of the areas of the two similar triangles using their side length ratio.

The ratio of the area of triangle

ABC

to the area of triangle

DEF

is

(3/5)^2 = 9/25

.

For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.

2

Set up a proportion to solve for the unknown area of triangle

DEF

.

\frac{18}{x} = \frac{9}{25}

, where

x

is the area of triangle

DEF

.

We equate the ratio of the actual areas to the theoretical ratio of areas derived from the side lengths.

3

Solve the proportion for

x

.

x = 18 \times \frac{25}{9} = 2 \times 25 = 50

.

Multiplying both sides by the reciprocal isolates the variable and yields the area.

Key Concept

The ratio of the areas of two similar triangles is the square of the ratio of their corresponding side lengths.

Question 596Question →

In a circle with center $O$ , segment $PT$ is tangent to the circle at point $T$ . The distance from point $P$ to the center of the circle is $25$ . If the radius of the circle is $7$ , what is the length of segment $PT$ ?

Show answer & explanation

Answer: 24

Answer

The length of segment

PT

is 24.

Because segment

PT

is tangent to the circle at point

T

, the radius

OT

is perpendicular to

PT

. This forms a right triangle

OTP

where the right angle is at vertex

T

, the legs are

OT = 7

and

PT

, and the hypotenuse is the segment from the center to the external point

OP = 25

. By the Pythagorean theorem,

OT^2 + PT^2 = OP^2

. Substituting the known lengths yields

7^2 + PT^2 = 25^2

, which simplifies to

49 + PT^2 = 625

. Subtracting

49

from both sides gives

PT^2 = 576

. Taking the square root of both sides results in

PT = 24

.

Step-by-Step Solution

1

Identify the relationship between the radius and the tangent line at the point of tangency.

The radius

OT

is perpendicular to the tangent segment

PT

, making triangle

OTP

a right triangle with a

90^\circ

angle at vertex

T

.

A tangent line to a circle is always perpendicular to the radius drawn to the point of tangency.

2

Set up the Pythagorean theorem for the right triangle

OTP

.

OT^2 + PT^2 = OP^2

In any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

3

Substitute the given values

OT = 7

and

OP = 25

into the equation and solve for the length of

PT

.

PT = 24

Substituting values gives

7^2 + PT^2 = 25^2 \implies 49 + PT^2 = 625 \implies PT^2 = 576 \implies PT = \sqrt{576} = 24

.

Key Concept

A line tangent to a circle is perpendicular to the radius at the point of tangency, allowing the use of the Pythagorean theorem to find unknown lengths in the resulting right triangle.

Question 597Question →

In a circle, chords $WY$ and $XZ$ intersect at point $P$ . The measure of minor arc $WX$ is $55^\circ$ and the measure of minor arc $YZ$ is $105^\circ$ . What is the measure, in degrees, of angle $WPX$ ?

Show answer & explanation

Answer: 80

Answer

80

The correct answer is 80. According to the intersecting chords angle theorem, when two chords intersect inside a circle, the measure of the angle they form is half the sum of the measures of the intercepted arcs. Here, angle

WPX

and its vertical angle intercept minor arcs

WX

and

YZ

. Therefore, the measure of angle

WPX

is

\frac{55^\circ + 105^\circ}{2} = \frac{160^\circ}{2} = 80^\circ

.

Step-by-Step Solution

1

Identify the geometric relationship for angles formed by intersecting chords inside a circle.

The measure of

\angle WPX

is equal to half the sum of the measures of its intercepted arc

WX

and the intercepted arc of its vertical angle, arc

YZ

.

By the intersecting chords angle theorem, the angle formed by two intersecting chords inside a circle is half the sum of the measures of the intercepted arcs.

2

Sum the measures of the intercepted arcs.

55^\circ + 105^\circ = 160^\circ

The measures of minor arcs

WX

and

YZ

are given as

55^\circ

and

105^\circ

respectively.

3

Divide the sum of the arc measures by 2.

80

Halving the sum of the arc measures (

160^\circ

) yields the measure of the angle:

\frac{160^\circ}{2} = 80^\circ

.

Key Concept

Intersecting Chords Angle Theorem

Question 598Question →

A designer has a rectangular piece of fabric that measures $12$ inches by $18$ inches. The designer cuts out two identical right triangular pieces from the corners along one of the $12$ -inch sides. Each right triangular piece has a leg of length $4$ inches along the $12$ -inch side and a leg of length $x$ inches along the $18$ -inch side. If the area of the remaining piece of fabric is $180$ square inches, what is the value of $x$ ?

Show answer & explanation

Answer: 9

Answer

9

The correct answer is 9. The original area of the rectangular fabric is

12 \times 18 = 216

square inches. Two identical right triangles with leg lengths of

4

inches and

x

inches are cut out. The area of each triangle is

\frac{1}{2} \times 4 \times x = 2x

square inches. The total area of the two triangles is

2 \times 2x = 4x

square inches. Subtracting this from the original area gives the remaining area:

216 - 4x = 180

. Solving for

x

gives

4x = 36

, which simplifies to

x = 9

.

Step-by-Step Solution

1

Calculate the area of the original rectangular piece of fabric.

216 square inches

To find the initial area before any modifications are made, using the formula

\text{Area} = \text{length} \times \text{width}

.

2

Find the total area of the two cut-out right triangles in terms of

x

.

4x

square inches

Each right triangle has legs of

4

and

x

, so its area is

\frac{1}{2}(4)(x) = 2x

. The total area of two such identical triangles is

2(2x) = 4x

.

3

Set up an equation using the remaining area of the fabric.

216 - 4x = 180

The remaining area of

180

square inches is equal to the original area of

216

square inches minus the total area of the two cut-out triangles, which is

4x

.

4

Solve the equation for

x

.

x = 9

Isolating the variable

x

by subtracting

216

from both sides and then dividing by

-4

yields

x = 9

.

Key Concept

Area of composite shapes (rectangles and triangles)

Question 599Question →

In a circle with center $O$ , chord $AB$ has a length of $12$ . The perpendicular distance from center $O$ to chord $AB$ is $8$ . If the area of the circle is $k\pi$ , what is the value of $k$ ?

Show answer & explanation

Answer: 100

Answer

100

A perpendicular from the center of a circle to a chord bisects that chord. For a chord of length

12

, the perpendicular split creates two segments of length

6

. Drawing a radius from the center to one of the chord's endpoints forms a right triangle with legs of

6

and

8

. By the Pythagorean theorem, the hypotenuse (which is the radius

r

) satisfies

r^2 = 6^2 + 8^2 = 100

. The area of the circle is

\pi r^2 = 100\pi

. Thus, the coefficient

k

is

100

.

Step-by-Step Solution

1

Determine the length of half of the chord.

6

A line segment drawn perpendicular from the center of a circle to a chord bisects the chord. Therefore, the distance from the midpoint of the chord to either endpoint is

12 / 2 = 6

.

2

Use the Pythagorean theorem to calculate the square of the radius.

r^2 = 100

The radius, half of the chord, and the perpendicular distance form a right-angled triangle. According to the Pythagorean theorem, the hypotenuse squared (

r^2

) is the sum of the squares of the legs:

r^2 = 6^2 + 8^2 = 36 + 64 = 100

.

3

Calculate the area of the circle in terms of

\pi

and identify the value of

k

.

k = 100

The formula for the area of a circle is

\pi r^2

. Since

r^2 = 100

, the area is

100\pi

. Comparing this to

k\pi

, we find that

k = 100

.

Key Concept

Perpendicular bisector of a circle chord and right triangle properties

Estimated Time:1m 30s

Question 600Question →

In right triangle $PQR$ , the measure of angle $Q$ is $90^\circ$ . If $\sin(P) = \frac{5}{13}$ and the length of side $QR$ is $15$ , what is the length of side $PQ$ ?

Show answer & explanation

Answer: 36

Answer

The length of side

PQ

is

36

.

By definition,

\sin(P) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{QR}{PR}

. Given that

\sin(P) = \frac{5}{13}

and

QR = 15

, we set up the equation

\frac{5}{13} = \frac{15}{PR}

and solve for the hypotenuse

PR

, giving

PR = 39

. Using the Pythagorean theorem,

PQ^2 + QR^2 = PR^2

, we substitute the known values:

PQ^2 + 15^2 = 39^2 \implies PQ^2 + 225 = 1521 \implies PQ^2 = 1296

. Taking the square root of both sides gives

PQ = 36

. Alternatively, recognizing that the sides of the triangle form a

5

-

12

-

13

Pythagorean triple scaled by a factor of

3

(since

QR = 5 \times 3 = 15

and

PR = 13 \times 3 = 39

), the remaining leg

PQ

must be

12 \times 3 = 36

.

Step-by-Step Solution

1

Set up the sine ratio for angle

P

to find the length of the hypotenuse

PR

.

PR = 39

Since

\sin(P)

is the ratio of the opposite side (

QR

) to the hypotenuse (

PR

), we can solve the equation

\frac{5}{13} = \frac{15}{PR}

to find that

PR = 39

.

2

Apply the Pythagorean theorem to solve for the length of side

PQ

.

PQ = 36

In right triangle

PQR

, the relationship between the sides is

PQ^2 + QR^2 = PR^2

. Substituting

QR = 15

and

PR = 39

gives

PQ^2 + 15^2 = 39^2

, which simplifies to

PQ^2 = 1296

, so

PQ = 36

.

Key Concept

Using trigonometric ratios to find side lengths of right triangles followed by the Pythagorean theorem.

Alternative Method

Recognize that the triangle's sides must be a multiple of the common

5

-

12

-

13

Pythagorean triple. Since the opposite side is

15

(

5 \times 3

) and the hypotenuse is

39

(

13 \times 3

), the scaling factor is

3

, meaning the adjacent side

PQ

is

12 \times 3 = 36

.

Estimated Time:1m 30s

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