Question

In $\Delta ABC,$ if $\cot \frac{A}{2}:\cot \frac{B}{2}:\cot \frac{C}{2}=3:7:9,$ then $a:b:c=$

• A. $7:9:11$
• B. $14:11:6$
• C. $7:19:2$
• D. $8:6:5$

Answer 1

The correct option is D $8:6:5$

As we know by using the half angle formula in sides form, The value of $\cot \left(\frac{A}{2}\right)=\sqrt{\frac{s(s-a)}{(s-b)(s-c)}}=\frac{s(s-a)}{\Delta }$

Since we know that, $\cot \frac{A}{2}:\cot \frac{B}{2}:\cot \frac{C}{2}=3:7:9,$

$\Rightarrow \Delta \cot \left(\frac{A}{2}\right):\Delta \cot \left(\frac{B}{2}\right):\Delta \cot \left(\frac{C}{2}\right)=3:7:9$

$\Rightarrow s-a:s-b:s-c=3:7:9$ As we know $s=\frac{a+b+c}{2}$

So putting value of $s$ and replacing ratio by constant $k$, we get $b+c-a=6k\cdots \left(i\right)$

$a+c-b=14k\cdots \left(ii\right)$

$a+b-c=18k\cdots \left(iii\right)$

Adding equation $\left(i\right)$ and equation $\left(ii\right),$ we get $2c=20k$

$\therefore c=10k$

Similarly, adding equation $\left(ii\right)$ and equation $\left(iii\right),$ we get $2a=32k$

$\therefore a=16k$

Similarly, adding equation $\left(i\right)$ and equation $\left(iii\right),$ we get $2b=24k$

$\therefore b=12k$

So, $a,b$ and $c$ are in the ratio of $16:12:10$, on dividing the ratio by $2$ we get, $a:b:c=8:6:5$.