Circles — Question 6

In figure,

A, D and S denote the positions of Ankur, David and Syed, respectively sitting at equal distance.

We know that,

If two minor arcs are equal in measure, then their corresponding chords are equal in measure.

∴ AD = DS = SA = 2x (let).

∴ ∆ ADS is an equilateral triangle.

From figure,

Radius = OA = OS = 20 m

Draw AB perpendicular to chord SD.

We know that,

Perpendicular from center bisects the chord.

∴ AB is the median.

∴ BS = BD = $\dfrac{SD}{2} = \dfrac{2x}{2}$ = x.

We know that,

Centre and Centroid are the same for an equilateral triangle, and it divides the median in the ratio 2 : 1.

∴ OA : OB = 2 : 1

$\Rightarrow \dfrac{20}{OB} = \dfrac{2}{1}$

$\Rightarrow OB = \dfrac{20}{2}$ = 10 m.

From figure,

AB = OA + OB = 20 + 10 = 30 m.

In right angle triangle ASB,

By pythagoras theorem,

⇒ AS² = AB² + BS²

⇒ (2x)² = 30² + x²

⇒ 4x² = 900 + x²

⇒ 4x² - x² = 900

⇒ 3x² = 900

⇒ x² = $\dfrac{900}{3}$

⇒ x² = 300

⇒ x = $\sqrt{300} = 10\sqrt{3}$

⇒ 2x = $2 \times 10\sqrt{3} = 20\sqrt{3}$.

Hence, the length of the string of each phone = $20\sqrt{3}$ m.