The angles of a triangle are in the ratio 5:1:6, then ratio of the smallest side to the greatest side is
The angles of a triangle are in the ratio 5:1:6, then ratio of the smallest side to the greatest side is
- $\sqrt{3}+1: 2 \sqrt{2}$
- $2 \sqrt{2}: \sqrt{3}+1$
- $2 \sqrt{2}: \sqrt{3}-1$
- $\sqrt{3}-1: 2 \sqrt{2}$
Solution
Let the angles of the triangle be $5 x, x, 6 x$
$\begin{array}{ll}
\therefore & 5 x+x+6 x=180^{\circ} \\
\therefore & 12 x=180^{\circ} \\
\therefore & x=15^{\circ}
\end{array}$
$\therefore \quad$ Three angles are $75^{\circ}, 15^{\circ}, 90^{\circ}$.
$\begin{aligned}
& \frac{\sin 75^{\circ}}{a}=\frac{\sin 15^{\circ}}{b}=\frac{\sin 90^{\circ}}{c}=k \\
\therefore \quad & \frac{b}{c}=\frac{\frac{\sin 15^{\circ}}{k}}{\frac{\sin 90^{\circ}}{k}}=\frac{\frac{\sqrt{3}-1}{2 \sqrt{2}}}{1}
\end{aligned}$
$\therefore \quad$ Required ratio $=\mathrm{b}: \mathrm{c}=\sqrt{3}-1: 2 \sqrt{2}$
Asked in: MHT CET 2024 (11 May Shift 2)
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