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Question

In a circle of radius $$\displaystyle 10.5 \ cm$$, the minor arc is one-fifth of the major arc. Find the area of the sector corresponding to the major arc.


Solution

Radius of circle =$$\displaystyle 10.5 \ cm$$
Let $$\displaystyle x \ cm$$ be the major arc, then $$\displaystyle \dfrac{x}{5} \ cm$$ be the length of minor arc.
Circumference of circle = $$\displaystyle x + \dfrac{x}{5} = \frac{6x}{5}$$
We know, Circumference of circle = $$\displaystyle 2\pi r = 2 \times \frac{22}{7} \times 10.5$$ 
This implies,
$$\displaystyle \frac{6x}{5} = 2 \times \frac{22}{7} \times 10.5$$
$$\displaystyle x = 55 \ cm$$
Area of major sector $$=\dfrac{\text{length of arc}}{2\pi r}\times\pi r^2=\displaystyle \frac{1}{2} \times 55 \times 10.5 = 288.75$$
The area of major sector is $$\displaystyle 288.75 \ cm^2$$.

Mathematics
RS Agarwal
Standard X

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