Question

In $△ABC$, if a: b: $c=4:5:6$, then the ratio of the circum radius to its inradius is

• A. $16:7$
• B. $7:16$
• C. $4:5$
• D. $5:4$

Answer 1

Answer: (A)

We have,
$a:b:c=4:5:6$
$\Rightarrow a=4K,b=5K,c=6K$
Now, $R=\frac{abc}{4\Delta }$ and $r=\frac{\Delta }{s}$
$\Rightarrow \frac{R}{r}=\frac{\left(\frac{abc}{4\Delta }\right)}{\left(\frac{\Delta }{s}\right)}=\frac{abcs}{4{\Delta }^2}$
Now, ${\Delta }^2=s(s-a)(s-b)(s-c)$ and
and
$s=\frac{a+b+c}{2}=\frac{4K+5K+6K}{2}=\frac{15K}{2}$
$\therefore \frac{R}{r}=\frac{\left[abc\frac{(a+b+c)}{2}\right]}{[4s(s-a)(s-b)(s-c)]}$
$=\frac{abc}{8(s-a)(s-b)(s-c)}$
$=\frac{4K\times 5K\times 6K}{4\left(\frac{15K}{2}-4K\right)\left(\frac{15K}{2}-5K\right)\left(\frac{15K}{2}-6K\right)}$
$=\frac{120K^3}{4\times \frac{7K}{2}\times \frac{5K}{2}\times \frac{3K}{2}}=\frac{16}{7}$
Required ratio $=16:7$