Question

In the following figure, $\Delta ABC$ is an isosceles triangles with perimeter $40cm$. The base $AC$ is of length $10cm$. Side $AB$ and side $BC$ are congruent. A circle touches the three sides as shown in the figure below. Find the length of the tangent segment from point $B$ to the circle.

Answer 1

$\begin{array}{c}Let,BP=BR=x.....\left(i\right)\\ AP=AQ=y.....\left(ii\right)\\ andCQ=CR=z......\left(iii\right)\end{array}\}$ (The length of two tangent segments to a circle from an external point are equal.)
AC = 10 cm (Given)
Perimeter of $\Delta ABC=40cm$ (Given)
$\therefore AB+BC+CA=40$
$(AP+BP)+(BR+CR)+(AQ+CQ)=40$
$\Rightarrow y+x+x+z+y+z=40$
$\Rightarrow 2(x+y+z)=40$
$\Rightarrow x+y+z=20$
$\Rightarrow x+(y+z)=20$
$\Rightarrow x+AC=20$
$\Rightarrow x+10=20$
$\Rightarrow x=20-10$
$\Rightarrow x=10$
$\Rightarrow BP=BR=10cm$
$\therefore$ Length of a tangent segment from point $B$ to the circle is $10cm$.