Question

In the given figure, \( PQ \) and \( PR \) are tangents to the circle such that \( PQ = 7 \, \text{cm} \) and \( \angle RPQ = 60^\circ \).

The length of chord QR is: 
 

• A. 5 cm
• B. 7 cm
• C. 9 cm
• D. 14 cm

Hint:

When tangents are drawn from an external point to a circle, they are equal in length. Use the cosine rule in triangle problems involving tangents and angle between them.

Answer 1

The Correct Option is D

Given that \( PQ = PR = 7 \, \text{cm} \) (tangents from an external point are equal) and \( \angle QPR = 60^\circ \), triangle \( \triangle PQR \) is an isosceles triangle with angle at vertex \( P = 60^\circ \).
Since the two sides are equal and the included angle is \( 60^\circ \), triangle \( PQR \) is also an equilateral triangle.
Therefore, all sides are equal:
\[ PQ = PR = QR = 7 \, \text{cm} \Rightarrow \text{But option (B) is 7 cm. Why not correct? Let's double-check:} \]
No, this is incorrect. The diagram clearly shows that both tangents \( PQ \) and \( PR \) are 7 cm, and angle between them is \( 60^\circ \). 
Now apply cosine rule to triangle \( \triangle PQR \):
\[ QR^2 = PQ^2 + PR^2 - 2 \cdot PQ \cdot PR \cdot \cos(\angle QPR) \]
\[ QR^2 = 7^2 + 7^2 - 2 \cdot 7 \cdot 7 \cdot \cos(60^\circ) = 49 + 49 - 98 \cdot \frac{1}{2} = 98 - 49 = 49 \Rightarrow QR = \sqrt{49} = 7 \, \text{cm} \]
So actual correct answer is (B) 7 cm, based on Cosine Rule, not (D). Let's correct the label:
Correct Answer: (B) 7 cm