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Byju's Answer

Standard IX

Mathematics

Theorem 1: Equal Chords Subtend Equal Angles at the Center

Two circles i...

Question

Two circles intersect at two points $B$ and $C$ . Through $B$ , two line segments $A B D$ and $P B Q$ are drawn to intersect the circles at $A, D$ and $P, Q$ respectively (see Fig.) Prove that $∠ A C P = ∠ Q C D$ .

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Solution

For chord $A P$
$∠ P B A = ∠ P C A$ (angle in same segment from common chord)
For chord $D Q$
$∠ D B Q = ∠ D C Q$ (angle in same segment from common chord)
Also,
$∠ P B A = ∠ D B Q$ (opposite angles)
Hence, $∠ P C A = ∠ D C Q = ∠ P B A = ∠ D B Q$
or $∠ A C P = ∠ Q C D$

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Similar questions

Q. Two circles intersect at two points

B

and

C

. Through

B

, two line segments

A B D

and

P B Q

are drawn to intersect the circles at

A, D

and

P, Q

respectively (see figure). Prove that

∠ A C P = ∠ Q C D

Q. Two circles intersects at two points

B

and

C

. Through

B

, two line segments

A B D

and

P B Q

are drawn to intersects the circles at

A, D

and

P, Q

respectively. Prove that

∠ A C P = ∠ Q C D

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 (see the given figure). Prove that $∠ A C P = ∠ Q C D$

Q. Two circles intersect at A and B. From a point P on one of these circles, two line segments PAC and PBD are drawn, intersecting the other circle at C and D respectively. Prove the CD is parallel to the tangent at P.

Q. In figure two circles intersect at two points P and Q. From a point A on a circle, two line segments APC and AQD are drawn intersecting the other circle at the points C and D. Prove that CD is parallel to the tangent at A.

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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 (see Fig.) Prove that ∠ACP=∠QCD.

For chord AP∠PBA=∠PCA (angle in same segment from common chord)For chord DQ∠DBQ=∠DCQ (angle in same segment from common chord)Also,∠PBA=∠DBQ (opposite angles)Hence, ∠PCA=∠DCQ=∠PBA=∠DBQor ∠ACP=∠QCD

Two circles intersect at two points $B$ and $C$ . Through $B$ , two line segments $A B D$ and $P B Q$ are drawn to intersect the circles at $A, D$ and $P, Q$ respectively (see Fig.) Prove that $∠ A C P = ∠ Q C D$ .

For chord $A P$
$∠ P B A = ∠ P C A$ (angle in same segment from common chord)
For chord $D Q$
$∠ D B Q = ∠ D C Q$ (angle in same segment from common chord)
Also,
$∠ P B A = ∠ D B Q$ (opposite angles)
Hence, $∠ P C A = ∠ D C Q = ∠ P B A = ∠ D B Q$
or $∠ A C P = ∠ Q C D$