In the adjoining figure,AB is the diameter of the circle of centre `O` & the chord CD is equal to radius. If P is an external point, then find the value of `angleAPB`.

Explanation

To find the value of angle \( APB \) in the given circle where \( AB \) is the diameter and \( CD \) is a chord equal to the radius, we can follow these steps:

### Step 1: Understand the Given Information
- \( AB \) is the diameter of the circle.
- \( O \) is the center of the circle.
- \( CD \) is a chord equal to the radius \( R \) of the circle.
- \( P \) is an external point.

### Step 2: Identify Key Properties
- The angle subtended by a chord at the center of the circle is twice the angle subtended at the circumference by the same chord.
- A diameter subtends a right angle to any point on the circumference.

### Step 3: Draw the Diagram
- Draw the circle with center \( O \), diameter \( AB \), and chord \( CD \).
- Mark point \( P \) outside the circle.

### Step 4: Analyze Triangle \( OCD \)
Since \( CD = R \) (the radius), and \( OC = OD = R \) (the radii of the circle), triangle \( OCD \) is an equilateral triangle. Thus:
- \( \angle COD = 60^\circ \).

### Step 5: Relate Angles Using Circle Properties
Using the property of angles subtended by a chord:
- \( \angle COD = 2 \times \angle CBD \)
- Therefore, \( 60^\circ = 2 \times \angle CBD \)
- This gives \( \angle CBD = 30^\circ \).

### Step 6: Use the Diameter Property
Since \( AB \) is the diameter, it subtends a right angle at point \( D \):
- \( \angle ADB = 90^\circ \).

### Step 7: Find Angle \( PDB \)
Since \( ADB \) and \( PDB \) form a linear pair:
- \( \angle ADB + \angle PDB = 180^\circ \)
- Thus, \( 90^\circ + \angle PDB = 180^\circ \)
- This leads to \( \angle PDB = 90^\circ \).

### Step 8: Analyze Triangle \( PDB \)
In triangle \( PDB \):
- The sum of angles is \( 180^\circ \):
- \( \angle PDB + \angle DBP + \angle DPB = 180^\circ \)
- Substituting the known values: \( 90^\circ + 30^\circ + \angle APB = 180^\circ \).

### Step 9: Solve for Angle \( APB \)
- Rearranging gives:
\[ \angle APB = 180^\circ - (90^\circ + 30^\circ) \]
\[ \angle APB = 180^\circ - 120^\circ \]
\[ \angle APB = 60^\circ \].

### Final Answer
Thus, the value of \( \angle APB \) is \( 60^\circ \).

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