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

Applying Sine rule,
$\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}=k$ where 'k' is any constant.
$⟹\frac{a}{sin60}=k,\frac{b}{sin90}=k,\frac{c}{sin30}=k⟹\frac{a}{\left(\frac{\sqrt{3}}{2}\right)}=k,\frac{b}{1}=k,\frac{c}{\left(\frac{1}{2}\right)}=k⟹\frac{2a}{\sqrt{3}}=k,\frac{b}{1}=k,\frac{2c}{1}=k⟹a=k\frac{\sqrt{3}}{2},b=kandc=\frac{k}{2}$
Given perimeter = $15+5\sqrt{3}$
$⟹a+b+c=15+5\sqrt{3}⟹k\frac{\sqrt{3}}{2}+k+\frac{k}{2}=15+5\sqrt{3}⟹3k+k\sqrt{3}=30+10\sqrt{3}$

Comparing each term, we get $k=10$. We know '$b$' which is the hypotenuse $=k$.
Hence the hypotenuse $=10cm$.