Tangents are drawn at extremities AB of a diameter of a circle centred at P. If tangents drawn at a point C on the circle meet the other two tangents respectively at Q and R then measure of `angle QPR` is

Options

A. a)`45^@`
B. b)`60^@`
C. c)`90^@`
D. d)`180^@`

Answer

To solve the problem, we need to find the measure of angle QPR formed by the tangents drawn from point C on the circle to points Q and R, where the tangents meet the tangents drawn at points A and B (the extremities of the diameter AB of the circle).

### Step-by-Step Solution:

1. **Understand the Circle and Points**:
- Let the center of the circle be P.
- A and B are the endpoints of the diameter of the circle.
- C is a point on the circumference of the circle.
- Tangents are drawn from point C to the circle, meeting the tangents at points Q and R.

2. **Identify the Properties of Tangents**:
- The lengths of tangents drawn from an external point to a circle are equal. Therefore, the lengths of tangents CQ and CR are equal (CQ = CR).

3. **Triangles Involved**:
- We will consider triangles PQR and PCB.
- In triangle PQR, we have:
- PQ = PR (since both are tangents from point C).
- PB = PC (since both are radii of the circle).

4. **Establish Similarity of Triangles**:
- Since PQ = PR and PB = PC, triangles PQR and PBC are similar by the Side-Side-Side (SSS) similarity criterion.

5. **Analyze Angles**:
- The angle at point P (angle QPR) can be analyzed using the properties of the circle.
- The angle APB (the angle subtended by the diameter at the circumference) is 180 degrees.
- Since angles in a triangle sum to 180 degrees, and considering the right angles formed by the tangents at points Q and R, we can conclude:
- Angle QPB = 90 degrees (tangent to radius).
- Angle PRB = 90 degrees (tangent to radius).

6. **Calculate Angle QPR**:
- Since angle APB = 180 degrees and angles QPB and PRB are both 90 degrees, we can find angle QPR:
- Angle QPR = 180 degrees - (Angle QPB + Angle PRB)
- Angle QPR = 180 degrees - (90 degrees + 90 degrees)
- Angle QPR = 180 degrees - 180 degrees = 0 degrees.
- However, this is not the case; we need to consider that angle QPR is actually half of angle APB due to the properties of tangents and circles.
- Therefore, angle QPR = 90 degrees.

### Final Answer:
The measure of angle QPR is **90 degrees**.