Question : The sides of a triangle are 24 cm, 26 cm and 10 cm. At each of its vertex, circles of radius 4.2 cm are drawn. What is the area (in cm2) of the portion covered by the three sectors of the circle? $\left(\pi=\frac{22}{7}\right)$
Option 1: 92.28
Option 2: 120
Option 3: 105.86
Option 4: 27.72
Correct Answer: 27.72
Solution :
The angles of three sectors are taken as $\theta_1, \theta_2$, and $\theta_3$.
The area of a sector of a circle with radius $r$ and angle $\theta$ is given by $\frac{1}{2} r^2 \theta$, where $\theta$ is in radians.
The sum of the areas of three sectors = $\frac{1}{2} r^2 (\theta_1 +\theta_2+\theta_3)$
Also, the sum of all the angles of a triangle is $\pi$.
So, the sum of the areas of three sectors = $\frac{1}{2} r^2 (\pi )$
= $\frac{1}{2} \times 4.2^2 \times \frac{22}{7}$
= 27.72 cm
2
Hence, the correct answer is 27.72.
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