Question

Three boys, Ankur, Syed and David are sitting at equal distances on the boundary of a circular park of radius $20m.$ Each of them has a toy telephone to talk to each other. Find the length of the string that connects the telephones of any two of the boys.

• A. 20√3
• B. 40√3
• C. 30√3
• D. 10√3

Answer 1

The correct option is A

20√3

Let Ankur, Syed and David be denoted by $A,S$ and $D$ respectively in the circular park.

$OS=OD=OA=20m$ [Radius $=20m$]

Construction: Draw $OM\perp SA$

Now, $AS=SD=AD$

So, $△ASD$ is an equilateral triangle.

$\angle A={60}^{\circ }$

$\angle MAO={30}^{\circ }$

In $△AMO$

$\sin {30}^{\circ }=\frac{MO}{AO}$

$\Rightarrow \frac{1}{2}=\frac{MO}{AO}$

$\Rightarrow MO=\frac{AO}{2}=\frac{20}{2}=10$

Applying Pythagoras Theorem in $△AMO$:

${AM}^2={AO}^2-{MO}^2={20}^2-{10}^2$

$=400-100=300$

$\Rightarrow AM=\sqrt{300}$

$\Rightarrow AS=2\times AM=2\sqrt{300}$

$=2\times 10\sqrt{3}=20\sqrt{3}m$