Question

Diameter AB and CD are perpendicular to each other. Radius is 7 cm.

Assertion: The perimeter of the shaded region is πr + 2r.

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:

In the given figure:

Diameter AB and CD are perpendicular and the circle is divided into four quarters.

The shaded region consists of two quarter circles (half the circle) placed opposite each other.

The perimeter of the shaded region is not the same as the perimeter of a semicircle.

The perimeter of the shaded region would be the sum of the curved parts of two quarter circles two arcs of length `(πr)/2` each plus the straight line segment joining their ends which is 2r.

So, the perimeter of the shaded region = `(πr)/2 + (πr)/2 + 2r = πr + 2r` only if the line segment joining these arcs is considered. But here, the line segment between quarter arcs which are perpendicular is effectively a diameter and not part of the shaded region boundary.

Actually, the perimeter of the shaded area is only the sum of the two quarter arcs, i.e., `(πr)/2 + (πr)/2 = πr`, since the line segments AB and CD are diameters split between shaded and unshaded parts, those diameters are not edges of the shaded area.

Therefore, the perimeter of the shaded region is not πr + 2r but only πr.

Thus:

Assertion (A) is false.

Reason (R) is true general perimeter of semicircle formula.

Hence, A is false but R is true.