In the figure , L is the centre of the circle, and ML is the perpendicular to LN. If the area of the triangel MLN is 36 then the area of the circle is:

Options

A. `68 pi`
B. `66 pi`
C. `70 pi`
D. `72 pi`

Answer

To solve the problem step by step, we will use the information given about the triangle MLN and the relationship between the triangle and the circle.

### Step 1: Understand the triangle and its area
Given that the area of triangle MLN is 36, we can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

### Step 2: Identify the base and height
In triangle MLN, we can consider:
- ML as the height (since ML is perpendicular to LN)
- LN as the base

Let the length of LN be \( b \) and the length of ML be \( h \). Therefore, we can write:
\[ \frac{1}{2} \times b \times h = 36 \]

### Step 3: Solve for \( b \times h \)
Multiplying both sides by 2, we get:
\[ b \times h = 72 \]

### Step 4: Relate the triangle to the circle
Since L is the center of the circle and ML is the height from M to LN, the length ML represents the radius of the circle (let's denote it as \( r \)). Thus, we can express the area of the circle using the formula:
\[ \text{Area of the circle} = \pi r^2 \]

### Step 5: Find the relationship between \( r \), \( b \), and \( h \)
From the right triangle MLN, we can use the Pythagorean theorem:
\[ LN^2 = ML^2 + MN^2 \]
This means that:
\[ b^2 = r^2 + MN^2 \]

However, we do not have enough information to find \( b \) and \( MN \) directly. Instead, we can use the area of the triangle to express \( r \).

### Step 6: Express \( r \) in terms of the area
Since we have \( b \times h = 72 \) and \( h = r \), we can express \( b \) as:
\[ b = \frac{72}{r} \]

### Step 7: Substitute \( b \) into the area of the circle formula
Now, we can express the area of the circle in terms of \( r \):
\[ \text{Area of the circle} = \pi r^2 \]

### Step 8: Calculate the area of the circle
Since we know that the area of triangle MLN is 36, and we have established the relationship, we can conclude that the radius \( r \) is related to the area of the triangle.

To find the area of the circle, we can assume a radius that satisfies the area condition. Since we have \( b \times h = 72 \) and \( h = r \), we can find \( r \) using the area of the triangle:
\[ \text{Area of the circle} = \pi r^2 = \pi \left( \frac{72}{b} \right)^2 \]

However, since we do not have a specific value for \( b \), we can conclude that the area of the circle is directly related to the area of the triangle, and we can assume a standard radius based on the area given.

### Final Calculation
Assuming \( r = 12 \) (since \( 36 = \frac{1}{2} \times 12 \times 12 \)), we can calculate the area of the circle:
\[ \text{Area of the circle} = \pi (12)^2 = 144\pi \]

### Conclusion
Thus, the area of the circle is:
\[ \text{Area of the circle} = 144\pi \]