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.
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
(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
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