A, B and C are three points on a circle. The tangent at C meets BA extended at T. Given, `angleATC = 36^@ and angle ACT = 48^@`, the angle subtended by AB at the centre of the circle is

Options

A. `84^@`
B. `48^@`
C. `96^@`
D. `72^@`

Answer

To solve the problem step by step, we will follow the geometric principles involved with angles in circles and tangents.

### Step-by-Step Solution:

1. **Draw the Circle and Points**:
- Draw a circle and mark three points A, B, and C on the circumference. Draw a tangent at point C that meets the line extended from AB at point T.

2. **Label the Angles**:
- According to the problem, we have:
- Angle ATC = 36°
- Angle ACT = 48°

3. **Use the Alternate Segment Theorem**:
- The Alternate Segment Theorem states that the angle subtended by a chord at the circumference is equal to the angle subtended by the same chord at the opposite segment. Therefore:
- Angle ABC = Angle ACT = 48°.

4. **Find the Exterior Angle**:
- In triangle ATC, angle BAC is the exterior angle. The exterior angle is equal to the sum of the two opposite interior angles:
- Angle BAC = Angle ATC + Angle ACT = 36° + 48° = 84°.

5. **Find Angle ACB**:
- Now, we need to find angle ACB. Since the angles in triangle ABC must sum up to 180°, we can find angle ACB:
- Angle ACB = 180° - Angle ABC - Angle BAC
- Angle ACB = 180° - 48° - 84° = 48°.

6. **Find the Angle at the Center**:
- The angle subtended by chord AB at the center (angle AOB) is twice the angle subtended at the circumference (angle ACB):
- Angle AOB = 2 × Angle ACB = 2 × 48° = 96°.

### Final Answer:
The angle subtended by AB at the center of the circle is **96°**.

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