Geometry Practice Test: Circle Arcs, Angles and Chords
- Circles: Arcs, Angles and Chords
- 15 questions
- 18 minutes
This test covers arcs and angles, arc length and sector area, tangent lines, segment lengths for chords and secants, and the equation of a circle. For angle questions, first locate the vertex: on the center, on the circle, inside it or outside it.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
Your result
- 0 correct
- 0 wrong
- 0 blank
-
Question 1
An inscribed angle of a circle intercepts an arc of 110°.
What is the measure of the inscribed angle?
Correct answer: A
An inscribed angle measures half of its intercepted arc: 110° ÷ 2 = 55°. A central angle, not an inscribed angle, would measure 110°.
-
Question 2
A central angle of a circle measures 72°.
What is the measure of the major arc determined by the angle?
Correct answer: D
The minor arc equals the central angle, 72°, and the whole circle is 360°, so the major arc is 360° − 72° = 288°. The value 108° is 180° − 72°, which is not an arc of this circle.
-
Question 3
A circle has a radius of 9 cm.
What is the length of the arc intercepted by a central angle of 80°?
Correct answer: B
Arc length is (80/360) · 2π · 9 = (2/9) · 18π = 4π cm. The value 18π cm is the whole circumference.
-
Question 4
A circle has a radius of 6 inches.
What is the area of a sector with a central angle of 150°?
Correct answer: C
Sector area is (150/360) · π · 6² = (5/12) · 36π = 15π in². Using the circumference instead of the area, (5/12) · 12π = 5π, gives an arc length, not an area.
-
Question 5
Triangle ABC is inscribed in a circle, and side AB is a diameter. m∠A = 34°.
What is m∠B?
Correct answer: B
An angle inscribed in a semicircle is a right angle, so ∠C = 90°. Then m∠B = 180° − 90° − 34° = 56°.
-
Question 6
Two chords intersect inside a circle. One of the angles formed intercepts an arc of 50°, and its vertical angle intercepts an arc of 86°.
What is the measure of the angle?
Correct answer: C
An angle formed by two chords that meet inside a circle is half the sum of the two intercepted arcs: (50° + 86°) ÷ 2 = 68°. Half the difference (18°) is the rule for angles whose vertex is outside the circle.
-
Question 7
Two secants are drawn to a circle from a point outside it. The intercepted arcs measure 130° (far arc) and 40° (near arc).
What is the measure of the angle formed by the secants?
Correct answer: A
For a vertex outside the circle, the angle is half the difference of the intercepted arcs: (130° − 40°) ÷ 2 = 45°. The value 85° is half the sum, which applies only when the vertex is inside the circle.
-
Question 8
Segment PA is tangent to circle O at point A. The radius is 5 and OP = 13.
What is the length of PA?
Correct answer: B
A tangent is perpendicular to the radius at the point of tangency, so △OAP has a right angle at A: PA² = 13² − 5² = 144 and PA = 12. The value 8 is just 13 − 5.
-
Question 9
From point P outside a circle, two segments are tangent to the circle at A and B. PA = 3x + 2 and PB = 5x − 8.
What is the length of PA?
Correct answer: C
Two tangent segments from the same outside point are congruent: 3x + 2 = 5x − 8, so x = 5 and PA = 3(5) + 2 = 17. The value 5 is only x.
-
Question 10
Chords AB and CD intersect at point E inside a circle. AE = 6, EB = 4 and CE = 3.
What is ED?
Correct answer: D
For intersecting chords, the products of the segment lengths are equal: AE · EB = CE · ED, so 24 = 3 · ED and ED = 8. The value 7 comes from adding (6 + 4 = 3 + 7) instead of multiplying.
-
Question 11
From point P outside a circle, tangent segment PT has length 6. A secant from P meets the circle first at Q and then at R, with PQ = 4.
What is the length of PR?
Correct answer: C
The tangent-secant rule says PT² = PQ · PR, so 36 = 4 · PR and PR = 9. The value 5 is the chord QR = 9 − 4, not the whole secant.
-
Question 12
Quadrilateral ABCD is inscribed in a circle. m∠A = (2x + 10)° and m∠C = (3x + 20)°.
What is m∠A?
Correct answer: B
Opposite angles of an inscribed quadrilateral are supplementary: 5x + 30 = 180, so x = 30 and m∠A = 2(30) + 10 = 70°. The value 110° is m∠C.
-
Question 13
Which equation represents a circle with center (3, −2) and radius 5?
Correct answer: A
The standard form is (x − h)² + (y − k)² = r². With center (3, −2) this gives (x − 3)² + (y + 2)², and the right side is 5² = 25, not 5.
-
Question 14
A circle has the equation x² + y² − 6x + 8y = 11.
What are its center and radius?
Correct answer: D
Completing the square gives (x − 3)² + (y + 4)² = 11 + 9 + 16 = 36. So the center is (3, −4) and the radius is 36 = 6; 36 is r², not r.
-
Question 15
A tangent and a chord meet at the point of tangency, and the arc inside the angle they form measures 140°.
What is the measure of the angle between the tangent and the chord?
Correct answer: A
An angle formed by a tangent and a chord at the point of tangency is half its intercepted arc: 140° ÷ 2 = 70°.
Questions you leave blank are not counted as wrong.
More Math tests
MathGeometry Practice Test: Area, Volume and Coordinate GeometryArea, Volume and Coordinate Geometry15 questions20 min MathGeometry Practice Test: Angles, Parallel Lines and Proof ReasoningAngles, Parallel Lines and Proof15 questions18 minMathGeometry Practice Test: Triangle Congruence and SimilarityTriangle Congruence and Similarity15 questions18 minMathGeometry Practice Test: Right Triangles and TrigonometryRight Triangles and Trigonometry15 questions20 minMathGeometry Practice Test: Parallel Lines and TransversalsParallel Lines and Transversals10 questions13 minMathGeometry Practice Test: Triangle Congruence (SSS, SAS, ASA, AAS, HL)Triangle Congruence10 questions13 minMathGeometry Practice Test: Similar TrianglesSimilar Triangles10 questions14 min