Question

An equilateral triangle circumscribes all the circles, each with radius $10cm$. What is the perimeter of the equilateral triangle?

• A. $20(2+\sqrt{3})cm$
• B. $60(2+\sqrt{3})cm$
• C. $30(2+\sqrt{3})cm$
• D. None of these

Answer 1

The correct option is B $60(2+\sqrt{3})cm$

As $ABC$ is an equilateral triangle $AC=AB=CB$
In right $△AOD,\frac{OD}{AD}=\tan {30}^{\circ }$
$AD=\frac{OD}{\tan {30}^{\circ }}=\frac{10}{\frac{1}{\sqrt{3}}}=10\sqrt{3}$
Similarly, $EC=10\sqrt{3}$
$AC=AD+DE+EC$
$=10\sqrt{3}+40+10\sqrt{3}$
$=40+20\sqrt{3}=20(2+\sqrt{3})$
Perimeter of triangle $=3\times AC$
$=3\times 20(2+\sqrt{3})=60(2+\sqrt{3})$.