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 = `(pirθ)/180`.

पर्याय

• 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 false but reason (R) is true.

Explanation:

Given:

• Original radius = r
• Original angle = θ

Perimeter of a sector = P = 2r + arc length

Arc length: `P = 2r + (pirtheta)/(180)`

New situation: 

Radius becomes `r/2`,
Angle becomes 2θ

New perimeter: 

`P' = 2(r/2) + (pi(r/2)(2theta))/(180)`

Simplify:

`P' = r + (pirtheta)/(180)`

Compare:

Original: `P = 2r + (pirtheta)/(180)`

New = `P' = r + (pirtheta)/(180)`

Clearly, P′ < P

So the perimeter decreases, not remains the same.