Question

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A chord of a circle of radius 12 cm subtends an angle of 120° at the centre. Find the area of the corresponding segment of the circle. (Take π= 3.14 and √3=1.73)

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Answer 1

hey mate
here's the solution

Answer 2

Answer:

88.44 $cm^2$

Step-by-step explanation:

Given :

Radius of circle = 12 cm

Angle ( θ ) = 120

We have to find area of the corresponding segment.

Let AO and OB are radius arc is making as APB

Now area of the corresponding segment = Area of sector - Area of Δ ABO.

First find the area of sector

$\large \text{Area of sector =\dfrac{\theta}{360} \pi \ r^2 }\\\\\\\large \text{Area of sector =\dfrac{120}{360} \pi \ 12^2 }\\\\\\\large \text{Area of sector =\dfrac{1}{3} \times3.14\times12\times12 \ cm^2 }\\\\\\\large \text{Area of sector =3.14\times12\times4 \ cm^2 }\\\\\\\large \text{Area of sector =150.72 \ cm^2 }$

Now  Area of Δ ABO.

Draw a perpendicular from O on AB. Perpendicular divide the chord we know.

Applying trigonometry ratio formula here

cos θ = B / H      and       sin θ = P / H

We are applying this to get base and height of Δ ABO.

$\large \sin \theta=\dfrac{base}{12}\\\\\\\large \text{Half angle of 120 = 60 and \sin60=\dfrac{\sqrt3}{2} }\\\\\\\large base=\dfrac{\sqrt3}{2}\times12 \ cm\\\\\\\large base=6\sqrt3 \ cm$

$\large \cos \theta=\dfrac{perpenducular}{12}\\\\\\\large \cos60=\dfrac{1}{2} \\\\\\\large perpenducular=\dfrac{1}{2}\times12 \ cm\\\\\\\large perpenducular=6 \ cm$

$\large real \ base=12\sqrt3 \ cm$

Now Area

$\large \text{Area of triangle =\dfrac{1}{2}\times real \ base \times perpendicular \ cm^2 }\\\\\\\large \text{Area of triangle =\dfrac{1}{2}\times12\sqrt3\times6 \ cm^2 }\\\\\\\large \text{Area of triangle =36\sqrt3 \ cm^2 }\\\\\\\large \text{Area of triangle =36\times1.73 \ cm^2 }\\\\\\\large \text{Area of triangle =62.28 \ cm^2 }$

Now Area of the corresponding segment = 150.72 - 62.28 $cm^2$

Area of the corresponding segment = 88.44 $cm^2$

Your answer is incorrect because you used wrong formula .

There should be sin ( θ / 2 ) × cos ( θ / 2 ) not 1 / 2.

Thus we get answer.