O is the centre of the circle of radius 6 cm and P is the mid -point of a chord AB .The length OP is 3 cm. The area (in sq cm ) of `/_\ AOB` is :

Options

A. `26pi`
B. `24pi`
C. `20pi`
D. `18pi`

Answer

To find the area of triangle AOB, we can follow these steps:

### Step 1: Understand the given information
- O is the center of the circle with a radius of 6 cm.
- P is the midpoint of the chord AB.
- The length OP is 3 cm.

### Step 2: Visualize the triangle
Draw a circle with center O. Mark the radius OA = OB = 6 cm. Draw the chord AB and mark its midpoint P. Connect OP, which is perpendicular to AB.

### Step 3: Use the properties of triangles
Since OP is perpendicular to AB, triangle AOB is divided into two right triangles, AOP and BOP. The height from O to the chord AB is OP = 3 cm.

### Step 4: Apply the Pythagorean theorem
In triangle AOP:
- OA (hypotenuse) = 6 cm (radius)
- OP (one side) = 3 cm (height)
- AP (the other side) = x (unknown)

Using the Pythagorean theorem:
\[ OA^2 = OP^2 + AP^2 \]
\[ 6^2 = 3^2 + x^2 \]
\[ 36 = 9 + x^2 \]
\[ x^2 = 36 - 9 \]
\[ x^2 = 27 \]
\[ x = \sqrt{27} = 3\sqrt{3} \text{ cm} \]

### Step 5: Calculate the area of triangle AOB
The area of triangle AOB can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

Here, the base AB = 2 * AP = 2x = 2(3\sqrt{3}) = 6\sqrt{3} cm, and the height OP = 3 cm.

Thus, the area of triangle AOB is:
\[ \text{Area} = \frac{1}{2} \times (6\sqrt{3}) \times 3 \]
\[ \text{Area} = \frac{1}{2} \times 18\sqrt{3} \]
\[ \text{Area} = 9\sqrt{3} \text{ cm}^2 \]

### Final Answer
The area of triangle AOB is \( 9\sqrt{3} \text{ cm}^2 \).

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