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CBSE Class 9 Mathematics Question 17 of 20

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.