Question

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

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

Answer 1

Answer: (D)

Let $a=13,b=14$ and $c=15$. Then
$S=\frac{a+b+c}{2}=\frac{13+14+15}{2}=21$
and area of triangle,
$\Delta =\sqrt{s(s-a)(s-b)(s-c)}$
$=\sqrt{21(21-13)(21-14)(21-15)}$
$=\sqrt{21\times 8\times 7\times 6}=7\times 3\times 4=84$
Now, as we know $R=\frac{abc}{4\Delta }$ and $r=\frac{\Delta }{s}$
$\therefore R=\frac{13\times 14\times 15}{4\times 84}$ and
$r=\frac{84}{21}$
$\Rightarrow R=\frac{65}{8}$ and $r=4$
So, $8R+r=65+4=69$