The areas of two sectors of two different circles with equal corresponding arc lengths are equal. Is this statement true? Why

Solution:

Take two circles of radii r₁, r₂ or arc length l₁, l₂ and θ₁, θ₂ as the corresponding angles of sectors

l₁ = r₁θ₁ π/180

l₂ = r₂θ₂ π/180

It is given that

l₁ = l₂

r₁θ₁ = r₂θ₂ = x

Take A1 and A2 as the area of sector

A₁ = πr₁²θ1/360

A₂ = πr2²θ2/360

Let us divide A₁ by A₂

\(\frac{A_{1}}{A_{2}}=\frac{\frac{\pi r_{1}\Theta _{1}r_{1}}{360^{0}}}{\frac{\pi r_{2}\Theta _{2}r_{2}}{360}}\) = xr1/xr2 = r1/r2

A₁/A₂ = r₁/r₂

Here

Area of sector can be equal when r₁/r₂ = 1

Area of sectors of two circles of same arcs length are not equal

Therefore, the statement is false.

✦ Try This: In a circle, an arc of length 4π cm subtends an angle of measure 40°

at the centre. Find the radius of the circle and the area of the sector corresponding to that arc.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 12

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NCERT Exemplar Class 10 Maths Exercise 11.2 Problem 9

The areas of two sectors of two different circles with equal corresponding arc lengths are equal. Is this statement true? Why

Summary:

The statement “The areas of two sectors of two different circles with equal corresponding arc lengths are equal” is false

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