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

False

Area of first sector = (1/2)(r₁)²θ₁ ,

where r₁ is the radius,

θ₁ is the angle in radians subtended by the arc at the center of the circle.

Area of second sector = (1/2)(r₂)²θ₂,

where r₂ is the radius,

θ₂ is the angle in radians subtended by the arc at the center of the circle.

Given that: (1/2)(r₁)²θ₁ = (1/2)(r₂)²θ₂

⇒ (r₁)²θ₁ = (r₂)²θ₂

It depends on both radius and angle subtended at the center. But arc length only depends on radius of the circle. Therefore, it is not necessary that the corresponding arc lengths are equal. It is possible only if corresponding angles are equal (because then, the corresponding radii will be equal and hence the arc lengths will be equal).