A B and C D are two chords of a circle intersecting at P. Prove that A P \times P B=C P \times P D.

ICSE Class X Math 2015 Solved Paper - Question 32

Question : 32 of 52

Marks: +1, -0

A B

and

C D

are two chords of a circle intersecting at

P

. Prove that

A P × P B=C P ×

P D

Solution:

To Prove:

{\AP} × {\PB}= {\CP} × {\PD}

Construction: Join

{\AD}

and CB.
Proof: In

△ {\APD}

and

△ {\CPB}

∠ {\A}=∠ {\C}

[angles of the same segment]

∠ {\C}=∠ {\B}

[angles of the same segment]

∴ △ {\APD} ∼ △ {\BPC}

[AA Postulate]

∴ \;\; \;{ {\AP}}/{ {\CP}}=\;{ {\PD}}/{ {\PB}}

[corresponding sides of similar

∆ s

are proportional]

∴ \;\; {\AP} × {\PB}= {\CP} × {\PD}

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