Circle Theorems and Arc Length Worksheet

Question 1

A circle has a radius of 6 cm. Find the length of an arc with central angle 60 degrees.

Accepted Answer

The arc length is 2π cm because s = (60/360) × 2π × 6 = 2π cm.

Question 2

In a circle, a central angle measures 120 degrees. What is the measure of its intercepted arc?

Accepted Answer

The intercepted arc measures 120 degrees because a central angle has the same measure as its intercepted arc.

Question 3

An inscribed angle intercepts an arc measuring 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

A diameter of a circle intercepts an arc. What is the measure of an inscribed angle that subtends the diameter?

Accepted Answer

The inscribed angle measures 90 degrees because any angle inscribed in a semicircle is a right angle.

Question 5

Two inscribed angles intercept the same arc. One angle measures 35 degrees. What is the measure of the other angle?

Accepted Answer

The other inscribed angle also measures 35 degrees because inscribed angles that intercept the same arc are congruent.

Question 6

A chord is 8 cm from the center of a circle with radius 10 cm. Find the length of the chord.

Accepted Answer

The chord length is 12 cm. Half of the chord forms a right triangle with legs 8 and x and hypotenuse 10, so x = 6. Therefore the full chord is 12 cm.

Question 7

In a circle, two congruent chords are given. What can you conclude about the arcs they intercept?

Accepted Answer

The intercepted arcs are congruent because congruent chords in the same circle intercept congruent arcs.

Question 8

A tangent and a radius meet at the point of tangency. What is the measure of the angle between them?

Accepted Answer

The angle measures 90 degrees because a tangent is perpendicular to the radius at the point of tangency.

Question 9

From the same exterior point, two tangent segments are drawn to a circle. One tangent segment is 11 cm long. Find the length of the other tangent segment.

Accepted Answer

The other tangent segment is 11 cm long because tangent segments drawn from the same exterior point are congruent.

Question 10

A circle has circumference 30π cm. Find the length of an arc whose measure is 144 degrees.

Accepted Answer

The arc length is 12π cm because s = (144/360) × 30π = 12π cm.

Question 11

Two chords intersect inside a circle. The intercepted arcs measure 110 degrees and 50 degrees. Find the measure of the angle formed by the chords.

Accepted Answer

The angle measures 80 degrees because an angle formed by two chords intersecting inside a circle equals half the sum of the intercepted arcs: (110 + 50) / 2 = 80.

Question 12

Two secants intersect outside a circle. The larger intercepted arc measures 160 degrees and the smaller intercepted arc measures 80 degrees. Find the exterior angle.

Accepted Answer

The exterior angle measures 40 degrees because an exterior angle formed by two secants equals half the difference of the intercepted arcs: (160 - 80) / 2 = 40.

Question 13

An inscribed quadrilateral has three angles measuring 88 degrees, 92 degrees, and 105 degrees. Find the fourth angle.

Accepted Answer

The fourth angle is 75 degrees because the angles of a quadrilateral sum to 360 degrees, and 360 - (88 + 92 + 105) = 75.

Question 14

A circle has radius 9 m. Find the arc length of a 200 degree arc. Give your answer in terms of π.

Accepted Answer

The arc length is 10π m because s = (200/360) × 2π × 9 = (5/9) × 18π = 10π m.

Question 15

In the same circle, arc AB measures 70 degrees and arc CD measures 70 degrees. What can you conclude about chords AB and CD?

Accepted Answer

Chords AB and CD are congruent because congruent arcs in the same circle subtend congruent chords.

Circle Theorems and Arc Length

Circle Theorems and Arc Length

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