Q. 175.0( 4 Votes )

In a circle, two chords AB and CD intersect at a point P inside the circle. Prove that

(a) ΔPAC ~ ΔPDB (b) PA • PB = PC • PD.


Answer :

Given: AB and CD are chords of the circle, intersecting at point P.

(a). To Prove: ∆PAC ∆PDB


Proof: In ∆PAC and ∆PDB,


APC = DPB [ they are vertically opposite angles]


CAP = PDB [ angles in the same segment are equal]


Thus, by AA-similarity criteria, we can say that,


∆PAC ∆PDB


Hence, proved.


(b). To Prove: PA × PB = PC × PD


Proof: As already proved that ∆PAC ∆PDB


We can write as,



By cross-multiplying, we get


PA × PB = PC × PD


Hence, proved.


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