Question

Assertion (A): If the radius of sector of a circle is reduced to its half and angle is doubled then the perimeter of the sector remains the same.

Reason (R): The length of the arc subtending angle θ at the centre of a circle of radius r = ${\displaystyle \frac{\pi r\theta }{180}}$.

Options

• Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

• Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A).

• Assertion (A) is true but reason (R) is false.

• Assertion (A) is false but reason (R) is true.

Answer 1

Assertion (A) is true but reason (R) is false.

Explanation:

Assertion (A):

The perimeter of a sector is given by: P = rθ + 2r

If the radius is reduced to half (r′ = r/2) and the angle is doubled (θ′ = 2θ), the new perimeter becomes:

P′ = r′θ′ + 2r′

Substitute r′ = r/2 and θ′ = 2θ:

${\displaystyle P\prime =\left(\frac{r}{2}\right)\left(2\theta \right)+2\left(\frac{r}{2}\right)}$

P′ = rθ + r

Reason (R):

Arc length for θ = 360^∘ = 2πr

If the angle subtended is some fraction θ/360^∘, the arc length becomes:

Arc length = ${\displaystyle \frac{\theta }{360°}\times 2\pi r=\frac{\pi r\theta }{180}}$