Chapter 10.2, Problem 12PSB

To determine

To Find : Length of perpendicular from centre.

Answer to Problem 12PSB

  $\text{Length of perpendicular is}\text{2}\sqrt{\text{3}}\text{ units}\text{.}$

Explanation of Solution

Given information : $\text{Regular hexagon with perimeter 24}\text{.}$

  

  $\begin{array}{l}\text{Perimeter = 24 }\Rightarrow \text{ 6 }\times \text{ side = 24}\\ \Rightarrow \text{Side = 24 unit, }\Rightarrow \text{ AB = unit}\\ \text{Clearly AB is chord of the circle}\\ \Rightarrow AP=PB\text{ [Perpendicular from centre to a chord bisects the chord]}\\ \Rightarrow AP=2\text{ unit}\\ \text{Let }\angle AOB=\theta \Rightarrow 6\theta =360°\Rightarrow \theta =60°\\ \Rightarrow \angle AOP=30°,[A\times OA=OB\text{ and }OP\perp AB]\\ \text{In }\Delta AOP,\\ \tan 30°=\frac{AP}{OP}\Rightarrow \frac{1}{\sqrt{3}}=\frac{2}{OP}\Rightarrow OP=2\sqrt{3}\text{ unit}\end{array}$