Question

The lengths of the sides of a triangle are $13,14$ and $15$. If $R$ and $r$ respectively denote the circum radius and in radius of that triangle, then $8R+r=$

• A. $84$
• B. $\frac{65}{8}$
• C. $4$
• D. $69$

Answer 1

The correct option is D $69$

Let $a,b$ and $c$ be lengths of sides of triangle.

Given, $a=13,b=14,c=15$

Thus $s=\frac{a+b+c}{2}=\frac{13+14+15}{2}=21$

Area of triangle $=\sqrt{s(s-a)(s-b)(s-c)}$

$=\sqrt{21\left(8\right)\left(7\right)\left(6\right)}$

Therefore, $\Delta =84;r=\frac{\Delta }{s}=4$

$R=\frac{abc}{4\Delta }=\frac{13\times 14\times 15}{84}=\frac{65}{2}$

Thus $8R+r=2\times \frac{65}{2}+4=69$