CIRCLES REVISION QUESTIONS

1. In a circle, the angle at the center is 120°. What is the angle at the circumference subtended by the same arc?
2. Two radii make an angle of 60° at the center. Find the angle subtended at the circumference by the same arc.
3. A triangle is inscribed in a semicircle. Prove the angle opposite the diameter is a right angle.
4. Chord AB and chord CD intersect at point E inside a circle. If AE = 4 cm, EB = 6 cm, and CE = 3 cm, find ED.
5. If two chords are equidistant from the center, what can be said about their lengths?
6. In a cyclic quadrilateral, one angle is 65°. Find its opposite angle.
7. Two chords AB and CD intersect outside the circle. If the external segments are 5 cm and 6 cm, and the total lengths are 11 cm and 9 cm, find the lengths of the unknown segments.
8. Find the angle subtended by an arc of 90° at the circumference of a circle.
9. In a circle, AB and CD are chords such that AB = CD and are equidistant from the center. Prove they are equal in length.
10. Prove that the perpendicular from the center of a circle to a chord bisects the chord.
11. If a radius bisects a chord that is not a diameter, prove it is perpendicular to the chord.
12. The angle subtended by a diameter at the circumference is 90°. Prove this using triangle properties.
13. In a circle, two chords AB and CD intersect at right angles at point E. If AE = 3 cm, EB = 2 cm, and CE = 4 cm, find ED.
14. In a cyclic quadrilateral, prove that opposite angles add up to 180°.
15. Chord PQ is 12 cm long and is 5 cm away from the center. Find the radius of the circle.
16. If O is the center of the circle and ∠AOB = 60°, find ∠ACB where C lies on the arc AB.
17. If arc AB subtends 50° at the center, what angle does it subtend at the circumference?
18. If the radius of the circle is 10 cm and the chord is 12 cm long, find the distance from the chord to the center.
19. Show that angles in the same segment are equal.
20. In triangle ABC inscribed in a circle, ∠B = 60°, ∠C = 70°. Find ∠A.
21. In a circle, chords AB and CD are such that AB = CD. Prove that the arcs subtended are equal.
22. In a cyclic quadrilateral, one angle is 85°. What is its opposite angle?
23. If two chords AB and CD are perpendicular bisectors of each other, prove that the intersection point is the center of the circle.
24. If ∠APB and ∠AQB are subtended by the same chord AB on the same side of the circle, prove they are equal.
25. In a circle, a chord of 8 cm is 3 cm from the center. Find the radius.
26. A line from the center of a circle is perpendicular to a chord. What can be concluded about the chord?
27. If the angle at the center is twice the angle at the circumference, prove this using triangle and isosceles triangle properties.
28. In triangle ABC inscribed in a circle, the angle at B is 90°. What side is the diameter?
29. Prove that the angle in a semicircle is always a right angle.
30. In a circle, AB and CD are chords such that AB = CD and ∠A = ∠C. Prove arc AB = arc CD.

Answers

1. 60°
2. 30°
3. 90° (Angle in a semicircle is a right angle)
4. ED = 8 cm
5. They are equal
6. 115°
7. Use: (external × total) = (external × total)
8. 45°
9. AB = CD
10. Yes, by perpendicular bisector theorem
11. Proven using triangle congruency
12. Using triangle formed with diameter as hypotenuse
13. ED = 6 cm
14. Opposite angles = 180°
15. Use: r² = x² + (d/2)² → r ≈ 6.5 cm
16. ∠ACB = 30°
17. 25°
18. Use Pythagoras → distance ≈ 8 cm
19. By arc subtending equal angles
20. ∠A = 50°
21. Equal arcs
22. 95°
23. Only if they intersect at 90°, they meet at the center
24. Equal angles subtended by same chord on same side
25. Use: r² = 3² + 4² = 25 ⇒ r = 5 cm
26. The chord is bisected
27. Triangle with vertex at center proves 2∠ at circumference
28. AC is the diameter
29. Proven geometrically
30. Equal chords subtend equal arcs