The radius of a circle is given as 15 cm and chord AB subtends an angle of `131^(@)` at the centre C of the circle . Using trigonometry. calculate :
the length of AB,

Answer

To find the length of chord AB in a circle with a radius of 15 cm that subtends an angle of 131 degrees at the center, we can follow these steps:

### Step 1: Understand the Geometry
- Draw a circle with center C and radius 15 cm.
- Mark points A and B on the circumference such that chord AB subtends an angle of 131 degrees at the center C.

### Step 2: Identify Key Points
- Since CA and CB are both radii of the circle, we have:
- CA = 15 cm
- CB = 15 cm
- The angle ∠ACB = 131 degrees.

### Step 3: Bisect the Angle
- Draw a perpendicular line from point C to chord AB, meeting it at point P. This line bisects chord AB into two equal segments, AP and PB.
- Therefore, AP = PB = AB/2.
- The angle ∠APC = ∠BPC = 131/2 = 65.5 degrees.

### Step 4: Use Trigonometry
- In triangle ACP, we can use the sine function:
\[ \sin(65.5^\circ) = \frac{AP}{AC} \]
where AC is the radius (15 cm).

### Step 5: Solve for AP
- Rearranging the equation gives:
\[ AP = AC \cdot \sin(65.5^\circ) \]
Substituting the values:
\[ AP = 15 \cdot \sin(65.5^\circ) \]
- Using the value of \(\sin(65.5^\circ) \approx 0.91\):
\[ AP = 15 \cdot 0.91 \approx 13.65 \text{ cm} \]

### Step 6: Find the Length of AB
- Since AP is half of AB:
\[ AB = 2 \cdot AP = 2 \cdot 13.65 \approx 27.30 \text{ cm} \]

### Final Answer
The length of chord AB is approximately **27.30 cm**.

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