Question

Assertion: AB is the diameter of the large semicircle with centre o and radius R. Two smaller semi-circles are drawn on AO and OB. The perimeter of the shaded part is 27πR.

Reason: The perimeter of a semi-circle is πR + 2R.

Options

• Both A and R are true and R is the correct reason for A.

• Both A and R are true but R is the incorrect reason for A.

• A is true but R is false.

• A is false but R is true.

Answer 1

A is false but R is true.

Explanation:

Assertion: The perimeter of the shaded part is 27πR

Let the total diameter AB = 2R, so radius of the large semicircle = R

Then:

• Arc of large semicircle = πR
• AO = OB = R, so radius of each small semicircle = `R/2`

So each small semicircle has arc:

`Arc = π xx R/2 xx 1/2`

`Arc = (πR)/4`

Two such arcs: `2 xx (πR)/4 = (πR)/2`

So, total perimeter of shaded region:

`πR + (πR)/2 = (3πR)/2`

This is clearly not 27πR.

So, Assertion is false.

Reason: The perimeter of a semicircle is πR + 2R.

Perimeter of a semicircle including straight diameter:

= Arc length + Diameter

= πR + 2R

So, Reason is true.