A circle is touching the side BC of a △ ABC at the point P and touching AB and AC produced at points Q and R…

CBSE Class 10 Maths PYQ · Circles · Triangle & Circle · 2 Marks · July 2023 · Standard

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842 Marks · July 2023 · Standard

A circle is touching the side BC of a $\triangle ABC$ at the point P and touching AB and AC produced at points Q and R respectively.
Prove that $AQ = \frac{1}{2}$ (Perimeter of $\triangle ABC$).

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Perimeter of $\triangle ABC = AB + BC + CA$
$= AB + BP + CP + CA$
$= AB + BQ + CR + CA$
$[BP = BQ \; ; \; CP = CR]$
$= AQ + AR$
$= AQ + AQ$
$[AQ = AR]$
$= 2 AQ$
$\therefore AQ = \frac{1}{2}$ (Perimeter of $\triangle ABC$)