Question

A circle is inscribed in a triangle whose sides are 8 cm , 15 cm , and 17 cm, then the radius of the circle is

• A. $3$ cm
• B. $4$ cm
• C. $5$ cm
• D. none of these

Answer 1

The correct option is A $3$ cm

$\Rightarrow$ Let in $△ABC$, $AB=8cm,BC=15cm$ and $AC=17cm$.

$\Rightarrow$ Consider, $A{B}^2+B{C}^2=8^2+{15}^2=64+225=289$ ---- ( 1 )

$\Rightarrow$ $A{C}^2={17}^2=289$ ------ ( 2 )

$\therefore$ $A{C}^2=A{B}^2+B{C}^2$ [ From ( 1 ) and ( 2 ) ]

$\Rightarrow$ Hence, $△ABC$ is right angled triangle.

$\therefore$ Area of $△ABC=\frac{1}{2}\times 8\times 15$

$\therefore$ Area of $△ABC=60c{m}^2$ ----- ( 3 )

$\Rightarrow$ Let $r$ be the radius of the incircle whose center is $O$.

$\Rightarrow$ Now, $ar(△AOB+△BOC+△COA$

$\Rightarrow$ $[\frac{1}{2}\times AB\times r]+[\frac{1}{2}\times BC\times r]+[\frac{1}{2}\times AC\times r]$

$\Rightarrow$ $\frac{1}{2}\times r\times (8+15+17)$

$\Rightarrow$ $\frac{1}{2}\times r\times 40$

$\Rightarrow$ $20r$

$\therefore$ $ar(△AOB+△BOC+△COA)=20r$ --- ( 4 )

$\Rightarrow$ But, $ar(△AOB+△BOC+△COA=ar(△ABC)$

$\therefore$ $20r=60$ [ From ( 3 ) and ( 4 ) ]

$\therefore$ $r=3cm$