Question

Assertion (A): If PA and PB are tangents to the circle centred at O and ∠OPA = 30° then ΔPAB is an equilateral triangle.

 

Reason (R): The lengths of tangents drawn from an external point to a circle are equal.

विकल्प

• Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

• Assertion (A) is true and Reason (R) is false.

• Assertion (A) is false and Reason (R) is true.

Answer 1

Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

Explanation:

Step 1 – Assertion: A is true. The two right triangles formed by joining O to the points of contact are congruent, so OP bisects the angle between the tangents. Hence, ∠APB = 60°. Also, PA = PB, so the base angles of triangle PAB are equal; each is 60°. Therefore, triangle PAB is equilateral.

Step 2 – Reason: R is true. Tangents drawn from the same external point to a circle have equal lengths.

Step 3 – Link: The tangent-length equality gives PA = PB and together with the given angle it establishes that all three angles of triangle PAB are 60°. Thus, R helps explain A.