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ACT · Math · Preparing for Higher Math: Geometry
Circles is part of Preparing for Higher Math: Geometry in the ACT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.
Question types you will see here include equation of a circle, center and radius from an expanded equation, circumference and area and arc length and sector area.
Common traps: using r instead of r squared; using the wrong sign for the center coordinates; mixing up the center and a point on the circle.
Question 1
Points A, B and C lie on a circle. Inscribed angle ABC measures 35°. What is the measure of arc AC, the arc intercepted by angle ABC?
17.5°
35°
70°
290°
Answer: C. 70°
An inscribed angle is half its intercepted arc, so the arc is 2 × 35° = 70°. A halves the angle instead of doubling it; B treats the inscribed angle as equal to its arc; D gives the other arc, 360° − 70°, which the angle does not intercept.
Question 2
In the standard (x, y) coordinate plane, a circle has the equation (x + 2)² + (y − 5)² = 49. What are the center and the radius of the circle?
Center (−2, 5); radius 7
Center (2, −5); radius 7
Center (−2, 5); radius 49
Center (2, −5); radius 49
Answer: A. Center (−2, 5); radius 7
Rewrite x + 2 as x − (−2), so h = −2 and k = 5, and r² = 49 gives r = 7. B takes the signs as printed instead of reversing them; C reports r² as the radius; D makes both mistakes.
Question 3
Segment AB is a diameter of a circle, and point C lies on the circle. Angle CAB measures 28°. What is the measure of angle ABC?
28°
56°
62°
152°
Answer: C. 62°
An angle inscribed in a semicircle is a right angle, so angle ACB = 90°. Then angle ABC = 180° − 90° − 28° = 62°. A assumes the two base angles are equal; B doubles 28° as if C were a central angle; D subtracts only 28° from 180°, ignoring the right angle.
Question 4
Quadrilateral ABCD is inscribed in a circle. Angle A measures (3x + 10)° and angle C, the angle opposite A, measures (2x + 20)°. What is the measure of angle A?
30°
40°
80°
100°
Answer: D. 100°
Opposite angles of an inscribed quadrilateral are supplementary: 5x + 30 = 180, so x = 30 and angle A = 100°. A reports x; B sets the two angles equal (x = 10, giving 40°); C is the measure of angle C, 2(30) + 20.
Question 5
A circle has a radius of 13. What is the length of an arc of the circle that has a central angle of 150°?
26π
(65/3)π
(845/12)π
(65/6)π
Answer: D. (65/6)π
Arc length = (θ/360)(2πr) = (150/360)(2π·13) = 65/6π.
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