Q. 185.0( 3 Votes )

Two chords AB and

Answer :

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

(a). To Prove: ∆PAC ∆PDB


Proof: We know


ABD + ACD = 180° [ opposite angles of cyclic quadrilateral are supplementary] …(i)


PCA + ACD = 180° [ they are linear pair angle] …(ii)


Comparing equations (i) & (ii), we get


ABD + ACD = PCA + ACD


ABD = PCA


Also, APC = BPD [ they are common angles]


Thus, by AA-similarity criteria, ∆PAC ∆PDB


Hence, proved.


(b). To Prove: PA × PB = PC × PD


Proof: We have already proved that, ∆PAC ∆PDB


Thus the ratios can be written as,



By cross-multiplication, we get


PA × PB = PC × PD


Hence, proved.


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