Question

AB is a chord of a circle with centre O and radius 4 cm. AB is of length 4 cm. Find the area of the sector of the circle formed by chord AB.

Answer 1

We have to find the area of the sector AOB formed by the chord AB. 

We have  ${\displaystyle OA=4cm\text{and}AB=4cm.So,}$

${\displaystyle AL=\frac{AB}{2}cm}$ 

${\displaystyle =\frac{4}{2}cm}$ 

${\displaystyle =2cm}$ 

Let ∠AOB=2θ. then, 

${\displaystyle \angle AOL=\angle BOL}$ 

=θ 

In ΔOLA, We have

${\displaystyle \sin \theta =\frac{AL}{OA}}$

${\displaystyle =\frac{2}{4}}$

${\displaystyle =\frac{1}{2}}$ 

${\displaystyle \theta ={\sin }^{-1} \frac{1}{2}}$ 

${\displaystyle =30°}$

Hence,  ${\displaystyle \angle AOB=60°}$

Now, using the value of${\displaystyle \angle AOB}$  and r we will find the area of sector AOB,

${\displaystyle A=\frac{\theta }{360°}\times \pi r^2}$ 

=${\displaystyle \frac{60°}{360°}\times \pi \times 4\times 4c{m}^2}$

${\displaystyle =\frac{8\pi }{3}c{m}^2}$

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