question_answer

P and Q are points on sides AB and AC respectively of \[\Delta ABC.\] If \[AP=3\,cm,\] \[PB=6\,cm,\] \[AQ=5\,cm\] and \[QC=10\,cm,\] then BC=

A)  PQ                             

B)  2PQ   

C)  3PQ   

D)         4PQ   

Correct Answer: C

Solution :

Since,  \[\frac{AP}{PB}=\frac{AQ}{QC}\] \[\therefore \]\[PQ||BC,\] [By converse of Thale's theorem] \[\Rightarrow \]\[\angle APQ=\angle ABC\] and \[\angle AQP=\angle ACB\] (Corresponding angles) \[\therefore \]   \[\Delta APQ\tilde{\ }\Delta ABC\]           [By AA similarity] \[\Rightarrow \]            \[\frac{AP}{AB}=\frac{PQ}{BC}=\frac{AQ}{AC}\] \[\Rightarrow \]            \[\frac{PQ}{BC}=\frac{1}{3}\]  \[\Rightarrow \]  \[BC=3PQ\]