Circles: Properties, Radians, and Applications

Question 1

Identify the equation of the circle with center at (2, -3) and radius 4:

Accepted Answer

$$(x - 2)^2 + (y + 3)^2 = 16$$

Answer

$$(x + 2)^2 + (y - 3)^2 = 16$$

Answer

$$(x + 2)^2 + (y + 3)^2 = 8$$

Answer

$$(x - 2)^2 + (y + 3)^2 = 8$$

Question 2

Determine the circumference of a circle with a radius of 7 cm:

Accepted Answer

14π cm

Answer

49π cm

Answer

28π cm

Answer

7π cm

Question 3

Determine the ratio of the area of a circle to the area of the square inscribed within the circle:

Accepted Answer

π:4

Answer

π:2

Answer

2:π

Answer

4:π

Question 4

What is the standard equation of a circle with center (h, k) and radius r?

Accepted Answer

(x - h)² + (y - k)² = r²

Answer

r(x + h) - (y - k) = 0

Answer

x² + y² - 2hx - 2ky + h² + k² = r²

Answer

(x - h)² - (y - k)² = r²

Question 5

If the diameter of a circle is 10 cm, what is its radius?

Accepted Answer

5 cm

Answer

10 cm

Answer

2,5 cm

Answer

20 cm

Question 6

A circle is inscribed in a square with side length 10 cm. What is the area of the circle?

Accepted Answer

25π square cm

Answer

50π square cm

Answer

100π square cm

Answer

10π square cm

Question 7

Calculate the circumference of a circle with a diameter of 12 cm.

Accepted Answer

12π or approximately 37.7 cm

Answer

24π cm

Answer

24 cm

Question 8

Determine the area of a circle with a radius of 5 meters.

Accepted Answer

25π m²

Answer

10π m²

Answer

50π m²

Question 9

Calculate the length of a circular arc with a radius of 10 cm and a central angle of 60 degrees.

Accepted Answer

(5π/3) cm

Answer

(3π/2) cm

Answer

(5π) cm

Answer

(π/3) cm

Question 10

Determine the area of a sector of a circle with a radius of 10 cm and a central angle of 60 degrees.

Accepted Answer

(25π/3) cm²

Answer

(50π) cm²

Answer

(5π/3) cm²

Answer

(20π) cm²

Question 11

What is the radius of a circle with the equation (x - 3)^2 + (y + 1)^2 = 25?

Accepted Answer

5

Answer

25

Answer

12

Answer

11

Question 12

Which of the following is a property of all circles?

Accepted Answer

They have a constant distance from a fixed point called the center.

Answer

They have parallel sides.

Answer

They always have an odd number of vertices.

Answer

They can be inscribed in a square.

Question 13

What is the measure of the central angle that intercepts an arc of length πr?

Accepted Answer

180 degrees

Answer

270 degrees

Answer

360 degrees

Answer

90 degrees

Question 14

Which of the following is a converse statement of the Pythagorean theorem?

Accepted Answer

If a triangle has sides of length a, b, and c, and a^2 + b^2 = c^2, then the triangle is a right triangle.

Answer

If a triangle has a right angle, then its sides satisfy the Pythagorean theorem.

Answer

If a^2 + b^2 = c^2, then the triangle with sides a, b, and c is isosceles.

Answer

If a triangle has two equal sides, then the Pythagorean theorem applies to it.

Question 15

What is the length of a chord that subtends a central angle of 90 degrees in a circle with radius 12 cm?

Accepted Answer

12√2 cm

Answer

12 cm

Answer

24 cm

Answer

6√2 cm

Question 16

Which of the following is the equation in standard form of a circle with center (0, 0) and radius 7?

Accepted Answer

x^2 + y^2 = 49

Answer

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

Answer

x^2 + y^2 = 7

Answer

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

Question 17

What is the measure of the inscribed angle that intercepts an arc of length πr?

Accepted Answer

90 degrees

Answer

120 degrees

Answer

45 degrees

Answer

180 degrees

Question 18

Which statement correctly expresses the relationship between the area and circumference of a circle?

Accepted Answer

Area = πr^2, Circumference = 2πr

Answer

Area = πd, Circumference = 2πd

Answer

Area = πr, Circumference = 2r

Answer

Area = 2πr^2, Circumference = πr

Question 19

Which of the following is the standard equation of a circle with center at the origin?

Accepted Answer

x^2 + y^2 = r^2

Answer

y = mx + b

Answer

x - y = r

Question 20

What is the measure of the circumference of a circle with radius 5 units?

Accepted Answer

10π

Answer

25π

Answer

5π

Question 21

Which of the following is a tangent to a circle?

Accepted Answer

A line that intersects the circle at exactly one point

Answer

A line that passes through the center of the circle

Answer

A line that is perpendicular to a radius of the circle

Question 22

What is the length of the radius of a circle that has a diameter of 8 units?

Accepted Answer

4

Answer

32

Answer

8

Answer

16

Question 23

Which of the following is a secant to a circle?

Accepted Answer

A line that intersects the circle at two points

Answer

A line that is tangent to the circle

Answer

A line that passes through the center of the circle

Answer

A line that is parallel to a chord of the circle

Question 24

What is the measure of the angle between two radii of a circle that intersect at a point other than the center?

Accepted Answer

Less than 180°

Answer

270°

Answer

180°

Answer

360°

Question 25

Determine the measure of the angle formed by two radii of a circle that are perpendicular to each other.

Accepted Answer

$$90°$$

Answer

$$45°$$

Answer

$$180°$$

Answer

$$270°$$

Question 26

Determine the measure of the angle formed by two chords of a circle that intersect inside the circle.

Accepted Answer

Half the sum of the measures of the intercepted arcs.

Answer

The sum of the measures of the intercepted arcs.

Answer

Half the difference of the measures of the intercepted arcs.

Question 27

What is the equation of a circle with center (h, k) and radius r?

Accepted Answer

(x - h)² + (y - k)² = r²

Answer

(x - h)² - (y - k)² = r²

Answer

(x + h)(y + k) = r

Answer

(x + h)² + (y + k)² = r

Question 28

Find the radius of a circle with circumference 20π cm.

Accepted Answer

10 cm

Answer

20 cm

Answer

5 cm

Answer

2 cm

Question 29

What is the area of a circle with diameter 14 cm?

Accepted Answer

154 cm² (rounded to two decimal places)

Answer

38.5 cm²

Answer

308 cm²

Answer

77 cm²

Question 30

What is the central angle that intercepts an arc of length 12 cm in a circle with radius 10 cm?

Accepted Answer

1.2 radians (rounded to two decimal places)

Answer

0.6 radians

Answer

1.8 radians

Answer

2.4 radians

Question 31

Which radian measure is equivalent to 90°?

Accepted Answer

π/2 radians

Answer

2π radians

Answer

π/4 radians

Answer

π radians

Question 32

Which of the following equations represents the circle with radius 5 and center at (2, -3)?

Accepted Answer

(x - 2)² + (y + 3)² = 25

Answer

(x + 2)² + (y - 3)² = 25

Answer

5x² + 5y² = 25

Answer

(x - 5)² + (y + 3)² = 25

Question 33

Which of the following is the equation that defines a circle?

Accepted Answer

x^2 + y^2 = r^2

Answer

y = mx + c

Answer

x^2 - y^2 = r^2

Answer

x + y = r

Question 34

If a circle has a diameter of 10 units, what is its radius?

Accepted Answer

5

Answer

10

Answer

50

Answer

25

Question 35

Which of the following angles is equivalent to one radian?

Accepted Answer

60°

Answer

180°

Answer

90°

Answer

30°

Question 36

What is the area of a circle with a radius of 5 units?

Accepted Answer

25π

Answer

50π

Answer

125π

Answer

5π

Question 37

Which of the following is considered a tangent to a circle?

Accepted Answer

A line that intersects the circle at exactly one point

Answer

A line that is parallel to the circle

Answer

A line that intersects the circle at two points

Answer

A line that passes through the center of the circle

Question 38

What is a chord of a circle?

Accepted Answer

A line segment that connects two points on the circle

Answer

A line that passes through the center of the circle

Answer

A line that intersects the circle at only one point

Question 39

On a circle with a radius of 10 cm, what is the measure of the inscribed angle that intercepts an arc of length 5 cm?

Accepted Answer

15°

Answer

30°

Answer

60°

Question 40

A circle has a radius of 5 units. What is its circumference?

Accepted Answer

10π units

Answer

None of the above

Answer

5π units

Answer

25π units

Question 41

If the center of a circle is at (2, 3) and its radius is 4, what are the coordinates of the circle?

Accepted Answer

(x - 2)^2 + (y - 3)^2 = 16

Answer

(x - 3)^2 + (y - 2)^2 = 16

Answer

(x + 3)^2 + (y + 2)^2 = 16

Answer

(x + 2)^2 + (y + 3)^2 = 16

Question 42

What is the relationship between the diameter and the radius of a circle?

Accepted Answer

Diameter = 2 * Radius

Answer

Radius = 2 * Diameter

Answer

Radius = Diameter^2

Answer

Diameter = Radius^2

Question 43

What is the measure of a central angle that intercepts an arc of 120 degrees?

Accepted Answer

120 degrees

Answer

360 degrees

Answer

180 degrees

Answer

60 degrees

Question 44

An arc of a circle measures 2π/3 radians. What is the measure of the corresponding central angle?

Accepted Answer

120 degrees

Answer

60 degrees

Answer

360 degrees

Answer

180 degrees

Question 45

A circle has a central angle of 60 degrees.  An arc intercepted by this angle has a length of 4π cm. What is the radius of the circle?

Accepted Answer

12 cm

Answer

24 cm

Answer

6 cm

Answer

3 cm

Question 46

Convert π/3 radians to degrees.

Accepted Answer

60°

Answer

120°

Answer

180°

Answer

30°

Question 47

What is the length of an arc that subtends a central angle of 2 radians in a circle with a radius of 5 units?

Accepted Answer

10 units

Answer

20 units

Answer

2.5 units

Answer

5 units

Question 48

A circle has an area of 16π square units. What is its circumference?

Accepted Answer

8π units

Answer

4π units

Answer

32π units

Answer

16π units

Question 49

Which of the following represents the standard equation of a circle with center (h, k) and radius r?

Accepted Answer

(x - h)² + (y - k)² = r²

Answer

(x + h)² + (y + k)² = r²

Answer

r² = (x - h)² + (y - k)²

Answer

r = (x - h)² + (y - k)²

Question 50

Find the equation of a circle that passes through the points (3, 4) and (7, 10).

Accepted Answer

(x - 5)² + (y - 7)² = 25

Answer

(x - 5)² + (y - 7)² = 50

Answer

(x - 3)² + (y - 4)² = 25

Answer

(x - 7)² + (y - 10)² = 25

Question 51

Find the circumference of a circle with radius 5 cm.

Accepted Answer

10π cm

Answer

5π cm

Answer

50π cm

Answer

25π cm

Question 52

Find the area of a circle with radius 6 cm.

Accepted Answer

36π cm²

Answer

12π cm²

Answer

72π cm²

Question 53

What is the measure of the central angle that intercepts an arc of length 4 cm on a circle with radius 5 cm?

Accepted Answer

48°

Answer

96°

Answer

72°

Answer

24°

Question 54

Find the area of the shaded region in the following diagram, where the circle has radius 6 cm and the rectangle has dimensions 8 cm by 4 cm.

Accepted Answer

24π - 32 cm²

Answer

16π - 32 cm²

Answer

48π - 32 cm²

Answer

32π - 16 cm²

Question 55

What is the value of sin(θ) for an angle θ that measures 60°?

Accepted Answer

√3/2

Answer

1/2

Answer

√2/2

Answer

1

Question 56

Find the measure of the angle x in the following diagram, where the circle has radius 10 cm.

Accepted Answer

30°

Answer

60°

Answer

15°

Answer

45°

Question 57

Which of the following is NOT a property of circles?

Accepted Answer

All circles are congruent.

Answer

Circles are symmetric figures.

Answer

Circles have a center point.

Answer

Circles can be inscribed in regular polygons.

Question 58

Find the equation of the circle with diameter AB, where A = (2, 5) and B = (6, 1).

Accepted Answer

(x - 4)² + (y - 3)² = 10

Answer

(x - 4)² + (y - 3)² = 20

Answer

(x - 2)² + (y - 5)² = 10

Answer

(x - 6)² + (y - 1)² = 10