Question

A circle of radius r is inscribed in a triangle of area ${}^{\prime }{\Delta }^{\prime }$. If the semi-perimeter of the triangle is s, then the correct relation is

• A. $2r=\frac{\Delta }{s}$
  
• B. $r=\frac{\Delta }{s}$
  
• C. $r=\frac{s}{\Delta }$
  
• D. $2s=\Delta r$

Answer 1

The correct option is B $r=\frac{\Delta }{s}$


We have, a circle of radius r is inscribed in a triangle of area ${}^{\prime }{\Delta }^{\prime }$

Here, AB, AC and BC are tangent to circle to circle and makes ${90}^{\circ }$ with radius

We know that,
Area of $\Delta ABC$ =Area of $\Delta OBA$ +Area of $\Delta OBC$ +Area of $\Delta OAC$
$=\frac{1}{2}\times r\times AB+\frac{1}{2}\times r\times BC+\frac{1}{2}\times r\times AC=\frac{1}{2}\times r(AB+BC+CA)=\frac{1}{2}\times r\times 2s\Rightarrow \Delta =rs\therefore r=\frac{\Delta }{s}$