Question

Assertion: In the semi circle, O is the centre. OPQR is a rectangle. OP = 6 cm, PQ = 7 cm. The length of PB = 11 cm.

Reason: The radius OQ = `sqrt(85)` cm.

विकल्प

• 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

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

Explanation:

Step 1: Evaluate the Reason (R):

We are given that OPQR is a rectangle with OP = 6 cm and PQ = 7 cm.

By properties of a rectangle, opposite sides are equal:

OR = PQ = 7 cm and QR = OP = 6 cm

O is the center of the semicircle, and Q is on the arc. OQ is the radius (r) of the semicircle.

Using the Pythagorean theorem in the right-angled triangle OQR:

OQ² = OR² + QR²

OQ² = 7² + 6²

OQ² = 49 + 36

OQ² = 85

OQ = `sqrt85` cm

∴ Reason (R) is true.

Step 2: Evaluate the Assertion (A):

The Assertion states that the length of PB = 11 cm.

Since O is the center and AOB is the diameter, OB is also a radius and equal to OQ:

OB = OQ = `sqrt85` cm

Using the Pythagorean theorem in the right-angled triangle OPB (O is the origin, P is at a height of 6 cm, B is at `(sqrt85, 0)`:

PB² = OP² + OB²

PB² = 6 + `(sqrt85)^2`

PB² = 36 + 85

PB² = 121

PB = `sqrt121`

PB = 11 cm

Assertion is true.

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