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  • The areas of two sectors of two different circles are equal. Is it necessary that their corresponding arc lengths are equal? Why?

The areas of two sectors of two different circles are equal. Is it necessary that their corresponding arc lengths are equal? Why?

Answers (1)

True

Solution

Let the radius of first circle is r1  and of other is r2

Let the arc length of both circles are same.

Let the arc length is a.                                 

length of arc (a)= 2\pi r \times \frac{\theta }{360^{\circ}}

Area of sector of first circle = a \times \frac{r_{1} }{2}       

(because area of sector = \pi r^{2}\times \frac{\theta }{360^{\circ}}=\left [ 2\pi r \times \frac{\theta }{360^{\circ}} \right ] \times \frac{r}{2}=\frac{r \times a}{2} )

Area of sector of other circle = a \times \frac{r_{2} }{2}

Here we found that both areas are equal in the case of when r1 = r2

Hence the area of two sectors of two different circles would be equal only in case of  both the circles have equal radii and equal corresponding arc length.

Hence it is necessary that their corresponding arcs lengths are equal.

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