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### Part (a): Special type of triangle \( \triangle ABT \)

Given the information:
- \( TA \) and \( TB \) are tangents to the circle from point \( T \).
- We know that the tangents from an external point to a circle are equal in length. Therefore, \( TA = TB \).
- This makes triangle \( \triangle ABT \) **isosceles**.

### Part (b): Calculate \( \angle AÔB \)

To calculate \( \angle AÔB \):
- \( \angle BÂT = 64^\circ \) is given.
- Since \( TA \) and \( TB \) are tangents, they make right angles with the radius of the circle at points \( A \) and \( B \) respectively. Thus, \( \angle OAT = \angle OBT = 90^\circ \).
  
Next, using the isosceles triangle property of \( \triangle AÔB \):
- The angle \( \angle AÔB \) is twice the angle subtended at the circumference by the same arc (since the angle at the center of a circle is twice the angle at the circumference). 
- Therefore, \( \angle AÔB = 2 \times \angle BÂT = 2 \times 64^\circ = 128^\circ \).

### Answers:
(a) **Isosceles triangle**

(b) \( \angle AÔB = 128^\circ \)

Let me know if you need further details or have any questions.

Here are five relative questions to deepen your understanding:
1. Why are the tangents from a single external point to a circle equal in length?
2. What are the properties of angles subtended by a chord at the circumference and the center of a circle?
3. How do the properties of tangents help in solving geometric problems involving circles?
4. Can \( \triangle ABT \) ever be a right triangle under different conditions?
5. How can you determine the angle between two tangents to a circle?

**Tip:** Always remember that angles subtended at the center of a circle by the same arc are twice the angles subtended at the circumference.

A and B are points on the circle, centre O. TA and TB are tangents to the circle. BÂT = 64°. (a) What special type of triangle is triangle ABT? (b) Work out ∠AÔB.

Solve a geometry problem involving tangents to a circle and the angle at the center. Explore the properties of tangents, isosceles triangles, and the angle subtended by the same arc. Suitable for Grades 9-11.

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