Question

A right triangle has angles which measure $30,60$ and $90$ degrees If the perimeter of this triangle is $15+5\sqrt{3}$ then the length of the hypotenuse of this triangle is

• A. $5$
• B. $7.5$
• C. $10$
• D. $12.5$

Answer 1

The correct option is C $10$

Let the three sides of the triangle be Perpendicular(P), Base (B) and Hypotenuse (H)
Now, $\sin 30=\frac{P}{H}$
$H=2P$ or $P=\frac{H}{2}$
$\cos 30=\frac{B}{H}$
$H=\frac{2B}{\sqrt{3}}$ or $B=\frac{\sqrt{3}H}{2}$
Perimeter of Triangle = $15+5\sqrt{3}$
$H+P+B=15+5\sqrt{3}$
$H+\frac{H}{2}+\frac{\sqrt{3}H}{2}=15+5\sqrt{3}$
$\frac{3H}{2}+\frac{\sqrt{3}H}{2}=15+5\sqrt{3}$
Thus, comparing the surds,
$3H=30$ or $H=10$