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 $R=\frac{a b c}{4 \Delta}$ 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, $ 8 R+r=65+4=69$