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CBSE Class 9 Mathematics Question 9 of 12

Circles — Question 9

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Question 9

Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively. Prove that ∠ACP = ∠QCD.

Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively. Prove that ∠ACP = ∠QCD. NCERT Class 9 Mathematics CBSE Solutions.
Answer

Join chord AP and DQ.

Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively. Prove that ∠ACP = ∠QCD. NCERT Class 9 Mathematics CBSE Solutions.

For chord AP,

⇒ ∠PBA = ∠ACP (Angles in the same segment are equal) .....(1)

For chord DQ,

⇒ ∠DBQ = ∠QCD (Angles in the same segment are equal) .....(2)

ABD and PBQ are line segments intersecting at B.

⇒ ∠PBA = ∠DBQ (Vertically opposite angles are equal) ....(3)

From equation (1) and (3) we get :

⇒ ∠DBQ = ∠ACP ...........(4)

From equation (2) and (4) we get :

⇒ ∠QCD = ∠ACP.

Hence, proved that ∠ACP = ∠QCD.

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Circles — Interactive Study Guide

Circle Theorems Quick Reference

  1. Equal chords ⇔ equal central angles
  2. Perpendicular from centre bisects the chord
  3. Equal chords ⇔ equidistant from centre
  4. Central angle = 2 × inscribed angle (same arc)
  5. Angles in same segment are equal
  6. Angle in semicircle = 90°
  7. Opposite angles of cyclic quad = 180°

Problem-Solving Toolkit

For chord problems: Drop perpendicular from centre → bisects chord → use Pythagoras.
For angle problems: Identify if angle is at centre or circumference → apply the 2x rule.
For cyclic quad: Opposite angles add up to 180°.

Quick Self-Check

  1. Inscribed angle = 35°. Central angle for the same arc? (70°)
  2. Chord = 10 cm, distance from centre = 12 cm. Radius? (√(25+144) = √169 = 13 cm)
  3. ABCD is cyclic, ∠A = 95°. Find ∠C. (85°)

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