Geometry Practice Test: Arcs and Chords of a Circle
- Circles: Arcs and Chords
- 10 questions
- 13 minutes
These questions focus on chords, the arcs they cut off and how they relate to the radius. Tip: draw the radius to the end of a chord; it often makes a right triangle with half the chord.
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Question 1
A chord of a circle is 16 cm long and is 6 cm from the center.
What is the radius of the circle?
Correct answer: C
The perpendicular from the center bisects the chord, forming a right triangle with legs 8 and 6. The radius is the hypotenuse: 64 + 36 = 10 cm.
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Question 2
A circle has a radius of 13 inches. A chord is 5 inches from the center.
How long is the chord?
Correct answer: A
Half the chord is 13² − 5² = 144 = 12 in., so the whole chord is 24 in. Answering 12 in. forgets to double.
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Question 3
In a circle, chords AB and CD are congruent. Minor arc AB measures (3x + 10)° and minor arc CD measures (5x − 20)°.
What is the measure of minor arc AB?
Correct answer: B
In the same circle, congruent chords cut off congruent minor arcs: 3x + 10 = 5x − 20, so x = 15 and the arc is 55°.
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Question 4
Points A, B and C lie on a circle. Minor arc AB measures 70°, minor arc BC measures 95°, and arcs AB and BC do not overlap.
What is the measure of arc AC that does not contain point B?
Correct answer: D
The arc from A to C through B is 70° + 95° = 165°. The rest of the circle, which does not contain B, is 360° − 165° = 195°.
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Question 5
AB is a diameter of a circle, and C is another point on the circle. AC = 6 and BC = 8.
What is the radius of the circle?
Correct answer: A
An angle inscribed in a semicircle is a right angle, so AB = 36 + 64 = 10. The radius is half the diameter: 5.
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Question 6
In circle O, chord PQ is 20 cm long and chord RS is 14 cm long.
Which statement is true?
Correct answer: B
Distance from the center is √(r² − (half chord)²), so a longer chord is closer to the center. PQ is longer, so it is closer.
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Question 7
Inscribed angles ∠ACB and ∠ADB both intercept arc AB. m∠ACB = (4x − 2)° and m∠ADB = (2x + 16)°.
What is the measure of arc AB?
Correct answer: C
Inscribed angles that intercept the same arc are congruent: 4x − 2 = 2x + 16 gives x = 9, so each angle is 34°. The arc is twice the inscribed angle: 68°.
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Question 8
Diameter CD is perpendicular to chord AB. Minor arc AB measures 110°, and D lies on minor arc AB.
What is the measure of arc AD?
Correct answer: A
A diameter perpendicular to a chord bisects the chord and its arc, so arc AD = 110°/2 = 55°. The value 125° is arc AC, on the other side.
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Question 9
In a circle with radius 9, a central angle of 60° cuts off a chord.
How long is the chord?
Correct answer: C
The two radii and the chord form an isosceles triangle with a 60° vertex angle, so all its angles are 60°. The triangle is equilateral and the chord equals the radius, 9.
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Question 10
In circle O, central angle ∠AOB measures (3x − 10)° and minor arc AB measures (2x + 20)°. Inscribed angle ∠ACB intercepts minor arc AB.
What is m∠ACB?
Correct answer: D
A central angle equals its arc: 3x − 10 = 2x + 20, so x = 30 and the arc is 80°. The inscribed angle is half the arc: 40°.
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