# A triangle ABC, right angled at B, is inscribed with a circle having 4cm diameter. The side BC is 5cm. What is the perimeter of the triangle?

Jul 18, 2018

Perimeter of the triangle is $30$ cm.

#### Explanation:

In right angle $\Delta A B C , A B , B C , A C$ are perpendicular (p),

base(b) and hypotenuse (h) respectively. The radius of the in-circle

is $r = \frac{4}{2} = 2$ cm , $B C = b = 5$ cm. In-radius of R.A.T is

$r = \frac{p + b - h}{2} \therefore \frac{p + 5 - h}{2} = 2 \mathmr{and} p + 5 - h = 4$

h= p+1 ; h^2= p^2+b^2 or h^2= p^2+25(Pythagoras formula).

or ${\left(p + 1\right)}^{2} = {p}^{2} + 25 \mathmr{and} \cancel{{p}^{2}} + 2 p + 1 = \cancel{{p}^{2}} + 25$

$\therefore 2 p = 24 \mathmr{and} p = 12 \therefore h = \sqrt{{12}^{2} + {5}^{2}} = 13$

$p = 12 , b = 5 , h = 13$. Perimeter of the triangle is

${T}_{p} = \left(p + b + h\right) = 12 + 5 + 13 = 30$ cm. [Ans]