Circle Geometry Practice Questions — SAT

Question 1Question →

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

Show answer & explanation

Answer: 21

Answer

21

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

O

perpendicular to both chords, we bisect chord

AB

at

M

(length 9) and chord

CD

at

N

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

AB

is

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

and to chord

CD

is

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

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

12 + 9 = 21

.

Step-by-Step Solution

1

Find the perpendicular distance from the center

O

to chord

AB

.

OM = 12

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

OMA

with hypotenuse

OA = 15

and leg

AM = 9

.

2

Find the perpendicular distance from the center

O

to chord

CD

.

ON = 9

Similarly, we use the Pythagorean theorem in right triangle

ONC

with hypotenuse

OC = 15

and leg

CN = 12

.

3

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

21

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

Key Concept

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

Question 2Question →

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

Show answer & explanation

Answer: 36

Answer

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

The total area of the circle is

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

. The shaded sector's area is

10\pi

, which represents

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

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

360^\circ

, the central angle of the sector is

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

.

Step-by-Step Solution

1

Calculate the total area of the circle.

Total area =

100\pi

The area of a circle is calculated using the formula

A = \pi r^2

where the radius

r = 10

.

2

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

Proportion =

\frac{1}{10}

Dividing the sector's area of

10\pi

by the total area of

100\pi

gives the fraction of the circle represented by the sector.

3

Find the central angle in degrees.

Central angle = 36

Since a full circle has a central angle of

360^\circ

, multiplying the proportion

\frac{1}{10}

by

360^\circ

yields the central angle of the sector.

Key Concept

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

360^\circ

).

Estimated Time:45s

Question 3Question →

A circle has a central angle of $60^\circ$ that subtends an arc of length $4\pi$ . What is the circumference of the circle?

12\pi

16\pi

24\pi

240\pi

Show answer & explanation

Answer: $24\pi$

Answer

The circumference of the circle is

24\pi

.

The correct answer is

24\pi

. A full circle is

360^\circ

. A central angle of

60^\circ

represents

\frac{60^\circ}{360^\circ} = \frac{1}{6}

of the entire circle. Therefore, the length of the arc subtended by this angle is

\frac{1}{6}

of the total circumference. To find the circumference, we multiply the arc length by

6

:

4\pi \times 6 = 24\pi

.

Step-by-Step Solution

1

Find the fraction of the circle represented by the

60^\circ

central angle.

\frac{60^\circ}{360^\circ} = \frac{1}{6}

A full circle has a central angle of

360^\circ

, so the fraction of the circle is the ratio of the given central angle to

360^\circ

.

2

Set up the proportion relating the arc length to the total circumference (

C

).

\frac{1}{6} = \frac{4\pi}{C}

The ratio of the arc length to the total circumference equals the ratio of the central angle to the total degree measure of a circle.

3

Solve for the circumference (

C

) by multiplying both sides by

6

.

C = 6 \times 4\pi = 24\pi

Multiplying the arc length of the sector by the reciprocal of the fraction yields the total circumference of the circle.

Key Concept

Finding the circumference of a circle given the central angle and the subtended arc length using proportional reasoning.

Question 4Question →

In a circle with center $O$ , central angle $AOB$ has a measure of $\frac{5\pi}{6}$ radians. If the radius of the circle is $12$ , the length of arc $AB$ is $k\pi$ . What is the value of $k$ ?

Show answer & explanation

Answer: 10

Answer

The value of

k

is

10

.

The arc length

s

subtended by a central angle

\theta

(in radians) in a circle of radius

r

is given by

s = r\theta

. Substituting the radius

r = 12

and the angle

\theta = \frac{5\pi}{6}

yields

s = 12 \times \frac{5\pi}{6} = 10\pi

. Since the arc length is expressed as

k\pi

, the value of

k

is

10

.

Step-by-Step Solution

1

Identify the formula for arc length in radians.

s = r\theta

The arc length

s

is directly proportional to the radius

r

and the central angle

\theta

in radians.

2

Substitute the given values into the formula.

s = 12 \times \frac{5\pi}{6}

The radius of the circle is

12

and the central angle is

\frac{5\pi}{6}

radians.

3

Calculate the arc length.

s = 10\pi

Simplifying

12 \times \frac{5}{6}

gives

2 \times 5 = 10

, so the product is

10\pi

.

4

Determine the value of

k

.

k = 10

We equate the calculated arc length

10\pi

with the given expression

k\pi

.

Key Concept

Arc length of a circle using radian measure

Estimated Time:45s

Question 5Question →

A circle has a circumference of $16\pi$ . An arc on this circle has a length of $2\pi$ . What is the measure, in degrees, of the central angle that subtends this arc?

22.5

40

45

90

Show answer & explanation

Answer: $45$

Answer

45

The correct answer is

45

because the ratio of the arc length to the total circumference is

\frac{2\pi}{16\pi} = \frac{1}{8}

. Since a full circle measures

360^\circ

, the central angle that subtends this arc is

\frac{1}{8} \times 360^\circ = 45^\circ

.

Step-by-Step Solution

1

Find the ratio of the arc length to the total circumference of the circle.

The ratio is

\frac{2\pi}{16\pi} = \frac{1}{8}

.

An arc length represents a fraction of the circle's total perimeter (circumference).

2

Multiply this fraction by the total degree measure of a circle (

360^\circ

) to find the measure of the central angle.

\frac{1}{8} \times 360^\circ = 45^\circ

.

The measure of the central angle is proportional to the fraction of the circle's circumference that the arc covers.

Key Concept

The relationship between arc length, circumference, and central angle measure in a circle.

Question 6Question →

In a circle with center $O$ , chord $AB$ has length $12\sqrt{3}$ . A radius $OC$ is perpendicular to chord $AB$ and intersects $AB$ at point $D$ . If $CD = 6$ , what is the area of the sector of the circle bounded by radii $OA$ , $OB$ , and the minor arc $AB$ ?

18\pi

24\pi

48\pi

144\pi

Show answer & explanation

Answer: $48\pi$

Answer

The correct answer is the option representing

48\pi

.

The correct answer is

48\pi

because the radius of the circle is determined to be

12

using the Pythagorean theorem, and the central angle of the sector is

120^\circ

. The area is then

\frac{120}{360} \times \pi (12)^2 = 48\pi

.

Step-by-Step Solution

1

Determine the length of the segment from the midpoint of the chord to its endpoints.

AD = 6\sqrt{3}

A radius perpendicular to a chord bisects the chord, so

D

is the midpoint of chord

AB

.

2

Express the distance from the center

O

to the chord intersection

D

in terms of the radius

R

and set up the Pythagorean theorem for right triangle

\triangle ODA

.

R^2 = (R - 6)^2 + (6\sqrt{3})^2

The radius

OC

has length

R

, making

OD = OC - CD = R - 6

. Since

\triangle ODA

is a right triangle, we can apply the Pythagorean theorem.

3

Solve the equation for the radius

R

.

R = 12

Expanding the equation gives

R^2 = R^2 - 12R + 36 + 108

, which simplifies to

12R = 144

.

4

Find the central angle

\angle AOB

.

\angle AOB = 120^\circ

In right triangle

\triangle ODA

, the cosine of

\angle AOD

is

\frac{OD}{OA} = \frac{6}{12} = \frac{1}{2}

, which means

\angle AOD = 60^\circ

. The total central angle is

\angle AOB = 2 \times \angle AOD = 120^\circ

.

5

Calculate the area of the sector bounded by

OA

,

OB

, and the minor arc

AB

.

48\pi

The area of the sector is the fraction of the circle's total area corresponding to the central angle:

\frac{120}{360} \times \pi (12)^2 = 48\pi

.

Key Concept

Using perpendicular bisector chord properties and right triangle trigonometry to determine circle sector area

Estimated Time:3m 0s

Question 7Question →

In a circle with center $O$ , $AB$ is a diameter. Point $C$ lies on the circle such that the measure of arc $AC$ is $\frac{5\pi}{9}$ radians. Point $D$ lies on the circle such that chord $AC$ is parallel to segment $OD$ , and points $C$ and $D$ lie on the same side of diameter $AB$ . What is the measure, in radians, of angle $COD$ ?

\frac{5\pi}{18}

\frac{2\pi}{9}

\frac{4\pi}{9}

\frac{13\pi}{18}

Show answer & explanation

Answer: $\frac{2\pi}{9}$

Answer

The correct answer is

\frac{2\pi}{9}

radians.

The correct answer is

\frac{2\pi}{9}

radians. First, the central angle

\angle AOC

has a measure of

\frac{5\pi}{9}

radians because it subtends an arc of the same measure. Since

OA

and

OC

are both radii of the circle, triangle

AOC

is isosceles with

OA = OC

, which means the base angles are equal:

\angle OAC = \angle OCA = \frac{\pi - 5\pi/9}{2} = \frac{2\pi}{9}

radians. Since chord

AC

is parallel to segment

OD

and the diameter

AB

acts as a transversal line, the corresponding angles

\angle OAC

and

\angle BOD

are equal, so

\angle BOD = \frac{2\pi}{9}

radians. Finally, because

A

,

O

, and

B

form a straight line, the angles along the diameter must sum to

\pi

radians:

\angle COD = \pi - \angle AOC - \angle BOD = \pi - \frac{5\pi}{9} - \frac{2\pi}{9} = \frac{2\pi}{9}

radians.

Step-by-Step Solution

1

Identify the measure of the central angle

\angle AOC

from the given arc measure.

The central angle

\angle AOC = \frac{5\pi}{9}

radians.

The measure of an arc in radians is equal to the measure of its subtended central angle.

2

Determine the measure of the inscribed angle

\angle OAC

using the properties of triangle

AOC

.

\angle OAC = \frac{2\pi}{9}

radians.

Since

OA

and

OC

are radii, triangle

AOC

is isosceles with

OA = OC

, meaning

\angle OAC = \angle OCA

. The sum of angles in a triangle is

\pi

radians, so

\angle OAC = \frac{\pi - \angle AOC}{2} = \frac{\pi - \frac{5\pi}{9}}{2} = \frac{2\pi}{9}

radians.

3

Use the parallel lines

AC

and

OD

to find the measure of angle

\angle BOD

.

\angle BOD = \frac{2\pi}{9}

radians.

Since chord

AC

is parallel to segment

OD

and diameter

AB

is a transversal line, the corresponding angles

\angle OAC

and

\angle BOD

are equal.

4

Calculate the measure of angle

COD

using the angles along the diameter

AB

.

\angle COD = \frac{2\pi}{9}

radians.

Points

A

,

O

, and

B

lie on a straight line, so the angles

\angle AOC

,

\angle COD

, and

\angle BOD

must sum to

\pi

radians. Therefore,

\angle COD = \pi - \angle AOC - \angle BOD = \pi - \frac{5\pi}{9} - \frac{2\pi}{9} = \frac{2\pi}{9}

radians.

Key Concept

Angle relationships in circles, including central angles, inscribed angles in isosceles triangles, and parallel line transversal properties.

Estimated Time:2m 30s

Question 8Question →

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 9Question →

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 10Question →

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 11Question →

Points $A$ , $B$ , and $C$ lie on a circle. The measure of minor arc $AB$ is $110^\circ$ and the measure of minor arc $BC$ is $130^\circ$ . What is the measure of the inscribed angle $\angle ABC$ , in degrees?

120

50

60

55

Show answer & explanation

Answer: 60

Answer

60

The measure of the major arc

ABC

is the sum of the minor arcs

AB

and

BC

, which is

110^\circ + 130^\circ = 240^\circ

. The remaining minor arc

AC

has a measure of

360^\circ - 240^\circ = 120^\circ

. By the inscribed angle theorem, the measure of the inscribed angle

\angle ABC

is half the measure of its intercepted arc, minor arc

AC

. Therefore, the measure of

\angle ABC

is

\frac{120^\circ}{2} = 60^\circ

.

Step-by-Step Solution

1

Calculate the measure of the major arc

ABC

by adding the measures of the two adjacent minor arcs

AB

and

BC

.

The measure of arc

ABC

is

110^\circ + 130^\circ = 240^\circ

.

Since points

A

,

B

, and

C

are in order on the circle, the major arc connecting

A

and

C

through

B

is the sum of the arcs

AB

and

BC

.

2

Find the measure of the remaining minor arc

AC

.

The measure of minor arc

AC

is

360^\circ - 240^\circ = 120^\circ

.

A full circle measures

360^\circ

. Subtracting the major arc

ABC

from

360^\circ

yields the measure of the minor arc

AC

.

3

Apply the inscribed angle theorem to find the measure of angle

ABC

.

The measure of angle

ABC

is

\frac{120^\circ}{2} = 60^\circ

.

The inscribed angle theorem states that the measure of an inscribed angle is half the measure of its intercepted arc. Angle

ABC

intercepts the minor arc

AC

.

Key Concept

The measure of an inscribed angle is half the measure of its intercepted arc.

Question 12Question →

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 13Question →

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 14Question →

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 15Question →

A circle with center $O$ has a radius of $8$ . Points $A$ and $B$ lie on the circle such that the length of the minor arc $AB$ is $5\pi$ . What is the area of the major sector $AOB$ ?

20\pi

36\pi

44\pi

11\pi

Show answer & explanation

Answer: $44\pi$

Answer

The area of the major sector is

44\pi

.

The area of the major sector is found by scaling the total area of the circle,

64\pi

, by the ratio of the major arc length to the circumference. The circumference is

2\pi(8) = 16\pi

. Since the minor arc length is

5\pi

, the major arc length is

16\pi - 5\pi = 11\pi

. The ratio of the major arc to the circumference is

\frac{11\pi}{16\pi} = \frac{11}{16}

. Multiplying this ratio by the total area of the circle yields

\frac{11}{16} \times 64\pi = 44\pi

.

Step-by-Step Solution

1

Calculate the circumference of the circle.

16\pi

The circumference formula is

C = 2\pi r

. Given

r = 8

, the circumference is

2\pi(8) = 16\pi

.

2

Find the length of the major arc

AB

.

11\pi

The length of the major arc is the total circumference minus the length of the minor arc:

16\pi - 5\pi = 11\pi

.

3

Calculate the total area of the circle.

64\pi

The area formula for a circle is

A = \pi r^2

. Given

r = 8

, the total area is

\pi(8^2) = 64\pi

.

4

Determine the area of the major sector by scaling the total area.

44\pi

The major sector's area is proportional to the fraction of the circle represented by the major arc:

\frac{11\pi}{16\pi} \times 64\pi = \frac{11}{16} \times 64\pi = 44\pi

.

Key Concept

Calculating sector area using arc length and total circle area relationships.

Estimated Time:1m 30s

Question 16Question →

In a circle with center $O$ , the length of minor arc $AB$ is $3\pi$ and the area of sector $AOB$ is $18\pi$ . What is the circumference of the circle?

12\pi

18\pi

24\pi

36\pi

Show answer & explanation

Answer: $24\pi$

Answer

The circumference of the circle is

24\pi

.

The correct answer is the option containing

24\pi

. By using the relationship

A = \frac{1}{2}rs

, where

A

is the sector area,

r

is the radius, and

s

is the arc length, we substitute the given values to get

18\pi = \frac{1}{2}r(3\pi)

. Solving for the radius yields

r = 12

. Substituting this radius into the circumference formula

C = 2\pi r

gives

2\pi(12) = 24\pi

.

Step-by-Step Solution

1

Write the formulas for arc length

s

and sector area

A

in terms of radius

r

and central angle

\theta

in radians.

s = r\theta = 3\pi

and

A = \frac{1}{2}r^2\theta = 18\pi

.

This sets up the system of equations using the given geometric properties.

2

Express the sector area formula in terms of arc length by substituting

s = r\theta

into

A = \frac{1}{2}r(r\theta)

.

A = \frac{1}{2}rs

, which becomes

18\pi = \frac{1}{2}r(3\pi)

.

This simplifies the relationship to a single equation with one variable,

r

.

3

Solve the equation

18\pi = 1.5\pi r

for the radius

r

.

r = 12

.

Finding the radius is necessary to calculate the circumference of the circle.

4

Substitute

r = 12

into the circumference formula

C = 2\pi r

.

C = 2\pi(12) = 24\pi

.

This provides the final circumference value requested by the question.

Key Concept

Relationship between arc length, sector area, and circumference in circle geometry

Question 17Question →

In a circle with center $O$ and radius $5$ , points $A$ , $B$ , and $C$ lie on the circle. If the measure of the inscribed angle $\angle ABC$ is $\frac{2\pi}{5}$ radians, what is the length of minor arc $AC$ ?

\pi

2\pi

4\pi

8\pi

Show answer & explanation

Answer: $4\pi$

Answer

4\pi

According to the Inscribed Angle Theorem, the measure of a central angle is twice the measure of an inscribed angle that subtends the same arc. Given that the inscribed angle

\angle ABC

has a measure of

\frac{2\pi}{5}

radians, the corresponding central angle

\angle AOC

has a measure of

2 \times \frac{2\pi}{5} = \frac{4\pi}{5}

radians. Using the formula for arc length,

s = r\theta

, where

r = 5

is the radius and

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

is the central angle in radians, the length of minor arc

AC

is

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

.

Step-by-Step Solution

1

Find the measure of the central angle

\angle AOC

that subtends the same minor arc

AC

as the inscribed angle

\angle ABC

.

The measure of central angle

\angle AOC

is

2 \times \frac{2\pi}{5} = \frac{4\pi}{5}

radians.

By the Inscribed Angle Theorem, the measure of a central angle subtending an arc is twice the measure of any inscribed angle subtending the same arc.

2

Calculate the length of minor arc

AC

using the formula

s = r\theta

.

The arc length is

s = 5 \times \frac{4\pi}{5} = 4\pi

.

The formula for the arc length of a circle is

s = r\theta

, where

r

is the radius and

\theta

is the central angle measure in radians.

Key Concept

Inscribed Angle Theorem and Arc Length in Radians

Question 18Question →

Points $A$ , $B$ , and $C$ lie on a circle with center $O$ . The length of the minor arc $AC$ is $\frac{4}{9}$ of the circumference of the circle. What is the measure, in degrees, of the inscribed angle $\angle ABC$ ?

Show answer & explanation

Answer: 80

Answer

The measure of the inscribed angle is 80 degrees.

The correct answer is 80. The minor arc

AC

constitutes

\frac{4}{9}

of the circle's circumference, which corresponds to an arc measure of

\frac{4}{9} \times 360^\circ = 160^\circ

. By the Inscribed Angle Theorem, the measure of the inscribed angle

\angle ABC

is half the measure of the intercepted arc, which is

\frac{1}{2} \times 160^\circ = 80^\circ

.

Step-by-Step Solution

1

Determine the degree measure of the minor arc

AC

.

The measure of minor arc

AC

is

160^\circ

.

Since a full circle has a circumference corresponding to

360^\circ

, minor arc

AC

has a degree measure of

\frac{4}{9} \times 360^\circ = 160^\circ

.

2

Calculate the measure of the inscribed angle

\angle ABC

.

The measure of

\angle ABC

is

80^\circ

.

According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of the arc it intercepts. Thus, the measure of

\angle ABC

is

\frac{1}{2} \times 160^\circ = 80^\circ

.

Key Concept

Inscribed Angle Theorem and Arc Measure

Question 19Question →

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

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