Circle Theorems and Arc Length
Using angles, chords, tangents, and arc measures in circles
Circle Theorems and Arc Length
Using angles, chords, tangents, and arc measures in circles
Math - Grade 9-12
- 1
A circle has a radius of 6 cm. Find the length of an arc with central angle 60 degrees.
Use the arc length formula with the fraction of the full circle.
The arc length is 2π cm because s = (60/360) × 2π × 6 = 2π cm. - 2
In a circle, a central angle measures 120 degrees. What is the measure of its intercepted arc?
The intercepted arc measures 120 degrees because a central angle has the same measure as its intercepted arc. - 3
An inscribed angle intercepts an arc measuring 86 degrees. Find the measure of the inscribed angle.
Inscribed angles are half of their intercepted arcs.
The inscribed angle measures 43 degrees because an inscribed angle is half the measure of its intercepted arc. - 4
A diameter of a circle intercepts an arc. What is the measure of an inscribed angle that subtends the diameter?
The inscribed angle measures 90 degrees because any angle inscribed in a semicircle is a right angle. - 5
Two inscribed angles intercept the same arc. One angle measures 35 degrees. What is the measure of the other angle?
Angles that intercept the same arc have the same measure.
The other inscribed angle also measures 35 degrees because inscribed angles that intercept the same arc are congruent. - 6
A chord is 8 cm from the center of a circle with radius 10 cm. Find the length of the chord.
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. - 7
In a circle, two congruent chords are given. What can you conclude about the arcs they intercept?
The intercepted arcs are congruent because congruent chords in the same circle intercept congruent arcs. - 8
A tangent and a radius meet at the point of tangency. What is the measure of the angle between them?
Recall the relationship between a radius and a tangent line.
The angle measures 90 degrees because a tangent is perpendicular to the radius at the point of tangency. - 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.
The other tangent segment is 11 cm long because tangent segments drawn from the same exterior point are congruent. - 10
A circle has circumference 30π cm. Find the length of an arc whose measure is 144 degrees.
Use the fraction of the circle represented by 144 degrees.
The arc length is 12π cm because s = (144/360) × 30π = 12π cm. - 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.
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. - 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.
For an exterior angle, subtract the smaller arc from the larger arc, then divide by 2.
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. - 13
An inscribed quadrilateral has three angles measuring 88 degrees, 92 degrees, and 105 degrees. Find the fourth angle.
The fourth angle is 75 degrees because the angles of a quadrilateral sum to 360 degrees, and 360 - (88 + 92 + 105) = 75. - 14
A circle has radius 9 m. Find the arc length of a 200 degree arc. Give your answer in terms of π.
Simplify the fraction before multiplying.
The arc length is 10π m because s = (200/360) × 2π × 9 = (5/9) × 18π = 10π m. - 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?
Chords AB and CD are congruent because congruent arcs in the same circle subtend congruent chords.