NIMCET 2015 — Mathematics PYQ

Let the angles of the triangle be $2x,3x,$ and $7x$.

Since the sum of angles in a triangle is $180^{\circ }$:

$2x+3x+7x=180^{\circ }$

$12x=180^{\circ }$

$x=15^{\circ }$

The angles are:

$A=2(15^{\circ })=30^{\circ }$

$B=3(15^{\circ })=45^{\circ }$

$C=7(15^{\circ })=105^{\circ }$

By Sine Rule, the ratio of sides $a:b:c$ is:

$a:b:c=\sin A:\sin B:\sin C$

$a:b:c=\sin 30^{\circ }:\sin 45^{\circ }:\sin 105^{\circ }$

Values:

$\sin 30^{\circ }=\frac{1}{2}$

$\sin 45^{\circ }=\frac{1}{\sqrt{2}}$

$\sin 105^{\circ }=\sin (60^{\circ }+45^{\circ })=\sin 60^{\circ }\cos 45^{\circ }+\cos 60^{\circ }\sin 45^{\circ }$

$\sin 105^{\circ }=\frac{\sqrt{3}}{2}\cdot \frac{1}{\sqrt{2}}+\frac{1}{2}\cdot \frac{1}{\sqrt{2}}=\frac{\sqrt{3}+1}{2\sqrt{2}}$

Ratio:

$a:b:c=\frac{1}{2}:\frac{1}{\sqrt{2}}:\frac{\sqrt{3}+1}{2\sqrt{2}}$

Multiply the entire ratio by $2\sqrt{2}$:

$a:b:c=\frac{2\sqrt{2}}{2}:\frac{2\sqrt{2}}{\sqrt{2}}:\frac{2\sqrt{2}(\sqrt{3}+1)}{2\sqrt{2}}$

$a:b:c=\sqrt{2}:2:\sqrt{3}+1$

Correct Option: (a)