Question

A circle touches the side BC of $\Delta$ ABC at P and touches AB and AC produced at Q and R respectively.
$IfAQ=5cm$, then find the perimeter of$\Delta$ ABC

• A. $5cm$
• B. $10cm$
• C. $25cm$
• D. $50cm$

Answer 1

Answer: (A)

$5cm$

$AQ=5$cm

Perimeter of $\Delta ABC$

$AQ=AR=5$cm

$\{$From a external point if drawn two tangents are equal in length$\}$

Perimeter of $\Delta ABC=AB+BC+AC$ .......$\left(1\right)$

But $BQ=BP$ .....$\left(2\right)$

$CP=CR$ ......$\left(3\right)$

$\{$From a external point if drawn tangents are equal in length$\}$

$\Rightarrow BP+PC=BQ+CR=BC$ ..........$\left(4\right)$

And $AQ=AB+BQ=AB+BP$ $\{$from equation $\left(2\right)\}$

$\Rightarrow AB=AQ-BP=5-BP$

$\Rightarrow AB+BP=5$

and $AR=AC+CR=AC+CP$ $\{$from equation $\left(3\right)\}$

$AC=AR-CP=5-CP$

$AC=CP=5$

From $\left(1\right)$

Perimeter$=AB+BC+AC$

$=AB+BP+PC+AC$ $\{$From equation $\left(4\right)\}$

$=5+5$

$=10$cm.