Geometry: Circles Worksheet

Question 1

A circle has radius 7 cm. Find its circumference and area. Leave both answers in terms of pi.

Accepted Answer

The circumference is 14π cm because C = 2πr = 2π(7). The area is 49π square cm because A = πr² = π(7²).

Question 2

In circle O, central angle AOB measures 128 degrees. What is the measure of minor arc AB?

Accepted Answer

The measure of minor arc AB is 128 degrees because the measure of a minor arc equals the measure of its central angle.

Question 3

An inscribed angle intercepts an arc that measures 86 degrees. Find the measure of the inscribed angle.

Accepted Answer

The inscribed angle measures 43 degrees because an inscribed angle is half the measure of its intercepted arc.

Question 4

Points A, B, C, and D lie on a circle. If angle ABC measures 52 degrees, what is the measure of arc AC that does not contain point B?

Accepted Answer

The arc AC that does not contain point B measures 104 degrees because an inscribed angle is half the measure of the arc it intercepts.

Question 5

A tangent line touches a circle at point T. Radius OT is drawn to the point of tangency. If angle OTP is formed by the radius and the tangent line, find its measure.

Accepted Answer

Angle OTP measures 90 degrees because a radius drawn to the point of tangency is perpendicular to the tangent line.

Question 6

From the same external point P, two tangent segments are drawn to a circle. The tangent segments are PA and PB. If PA = 3x + 4 and PB = 5x - 10, find x and the length of each tangent segment.

Accepted Answer

The value of x is 7 because tangent segments from the same external point are congruent, so 3x + 4 = 5x - 10. Each tangent segment is 25 units long because 3(7) + 4 = 25 and 5(7) - 10 = 25.

Question 7

A chord is 16 inches long and is 6 inches from the center of a circle. Find the radius of the circle.

Accepted Answer

The radius is 10 inches. The perpendicular from the center to a chord bisects the chord, so half the chord is 8 inches. Using the right triangle, r² = 6² + 8² = 36 + 64 = 100, so r = 10.

Question 8

Two chords intersect inside a circle. One chord is split into segments of lengths 4 and 15. The other chord is split into segments of lengths 6 and x. Find x.

Accepted Answer

The value of x is 10. By the intersecting chords theorem, 4 · 15 = 6 · x, so 60 = 6x and x = 10.

Question 9

Two secants are drawn from the same external point. The first secant has an external segment of 5 and a whole length of 20. The second secant has an external segment of 8 and a whole length of x. Find x.

Accepted Answer

The value of x is 12.5. By the secant-secant theorem, 5 · 20 = 8 · x, so 100 = 8x and x = 12.5.

Question 10

A tangent and a secant are drawn from the same external point. The tangent segment is 12 units long. The secant has an external segment of 8 units and a whole length of x units. Find x.

Accepted Answer

The whole secant length is 18 units. By the tangent-secant theorem, 12² = 8x, so 144 = 8x and x = 18.

Question 11

Find the length of a 60 degree arc in a circle with radius 9 cm. Leave your answer in terms of pi.

Accepted Answer

The arc length is 3π cm. Use s = (θ/360) · 2πr, so s = (60/360) · 2π(9) = (1/6) · 18π = 3π.

Question 12

Find the area of a sector with central angle 120 degrees in a circle with radius 6 meters. Leave your answer in terms of pi.

Accepted Answer

The sector area is 12π square meters. Use A = (θ/360) · πr², so A = (120/360) · π(6²) = (1/3) · 36π = 12π.

Question 13

Write the equation of a circle with center (3, -2) and radius 5.

Accepted Answer

The equation is (x - 3)² + (y + 2)² = 25. A circle with center (h, k) and radius r has equation (x - h)² + (y - k)² = r².

Question 14

Find the center and radius of the circle given by the equation (x + 4)² + (y - 1)² = 49.

Accepted Answer

The center is (-4, 1), and the radius is 7. In standard form, (x - h)² + (y - k)² = r², so h = -4, k = 1, and r = √49 = 7.

Question 15

Complete the square to write x² + y² - 6x + 8y - 11 = 0 in standard form. Then identify the center and radius.

Accepted Answer

The standard form is (x - 3)² + (y + 4)² = 36. The center is (3, -4), and the radius is 6. This comes from x² - 6x = (x - 3)² - 9 and y² + 8y = (y + 4)² - 16.

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Geometry: Circles

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