The tangents at two points A and B on the circle with centre O Intersects at P, If in quadrilateral `AOBP, /_AOB: /_APB= 5:1`, then measure of `/_APB `is:

Options

A. `30^@`
B. `60^@`
C. `45^@`
D. `15^@`

Answer

To solve the problem, we need to find the measure of angle APB in the quadrilateral AOBP, given that the ratio of angles AOB to APB is 5:1.

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have a circle with center O and points A and B on the circumference. The tangents at points A and B intersect at point P.
- The angles we are interested in are angle AOB and angle APB.

2. **Define the Angles**:
- Let angle AOB = 5x (as per the ratio given).
- Let angle APB = x.

3. **Use the Property of Tangents**:
- The radius of the circle is perpendicular to the tangent at the point of tangency. Therefore, angles OAP and OBP are both right angles (90 degrees).
- This gives us the following angles in quadrilateral AOBP:
- Angle OAP = 90°
- Angle OBP = 90°

4. **Set Up the Equation**:
- The sum of the angles in quadrilateral AOBP must equal 360 degrees:
\[ \text{Angle AOB} + \text{Angle OAP} + \text{Angle OBP} + \text{Angle APB} = 360° \]
- Substituting the known values:
\[ 5x + 90° + 90° + x = 360° \]

5. **Simplify the Equation**:
- Combine like terms:
\[ 5x + x + 180° = 360° \]
\[ 6x + 180° = 360° \]

6. **Solve for x**:
- Subtract 180° from both sides:
\[ 6x = 360° - 180° \]
\[ 6x = 180° \]
- Divide both sides by 6:
\[ x = 30° \]

7. **Find Angle APB**:
- Since angle APB = x, we have:
\[ \text{Angle APB} = 30° \]

### Final Answer:
The measure of angle APB is **30 degrees**.