A circle (with centre at O) is touching two intersecting lines AX and BY. The two points of contact A and B subtend an angle of `65^(@)` at any point C on the circumference of the circle. If P is the point of intersection for the two lines, then the measure of `angle APO` is :

Options

A. `25^(@)`
B. `65^(@)`
C. `90^(@)`
D. `40^(@)`

Answer

To solve the problem step by step, we will analyze the given information and apply the properties of circles and angles.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have a circle with center O that touches two intersecting lines AX and BY at points A and B, respectively.
- The lines AX and BY intersect at point P.
- The angle subtended by points A and B at any point C on the circumference of the circle is given as 65°.

2. **Finding Angle AOB**:
- The angle AOB is the angle at the center of the circle subtended by the arc AB.
- According to the property of circles, the angle at the center is twice the angle at the circumference.
- Therefore, \( \angle AOB = 2 \times \angle ACB = 2 \times 65° = 130° \).

3. **Analyzing Quadrilateral APBO**:
- In quadrilateral APBO, we know three angles:
- \( \angle OAP = 90° \) (since AX is tangent to the circle at A),
- \( \angle OBP = 90° \) (since BY is tangent to the circle at B),
- \( \angle AOB = 130° \) (calculated in the previous step).
- The sum of the angles in a quadrilateral is 360°.
- Therefore, we can write the equation:
\[ \angle OAP + \angle OBP + \angle AOB + \angle APB = 360° \]
Substituting the known values:
\[ 90° + 90° + 130° + \angle APB = 360° \]
\[ 310° + \angle APB = 360° \]
\[ \angle APB = 360° - 310° = 50° \]

4. **Finding Angles in Triangle APO**:
- In triangle APO and triangle BPO, we can see that:
- OA and OB are radii of the circle (equal in length),
- AP and BP are tangents from point P to the circle (equal in length).
- Therefore, triangles APO and BPO are congruent by the SSS (Side-Side-Side) criterion.
- This means \( \angle APO = \angle BPO \).

5. **Relating Angles APO and APB**:
- Since \( \angle APB = \angle APO + \angle BPO \) and both angles are equal, we can write:
\[ \angle APO + \angle APO = 50° \]
\[ 2 \times \angle APO = 50° \]
\[ \angle APO = \frac{50°}{2} = 25° \]

### Final Answer:
The measure of \( \angle APO \) is \( 25° \).

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