The length of three medians of a triangle are 9 cm, 12 cm and 15 cm. The area (in sq. cm) of the triangle is 

Options

A. 24
B. 72
C. 48
D. 144

Answer

To find the area of a triangle given the lengths of its medians, we can use the formula that relates the area of the triangle to its medians. The area \( A \) of the triangle can be calculated using the formula:

\[ A = \frac{4}{3} \times \text{Area of triangle formed by the medians} \]

### Step-by-step Solution:

1. **Identify the lengths of the medians**:
The lengths of the medians are given as 9 cm, 12 cm, and 15 cm.

2. **Calculate the area of the triangle formed by the medians**:
We can treat the lengths of the medians as the sides of a triangle. We will use the formula for the area of a triangle using the base and height. Since we have a right triangle formed by the medians, we can take two of the medians as the base and height.

Here, we can use 9 cm and 12 cm as the base and height respectively.

\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 9 \times 12 \]

3. **Calculate the area**:
\[ \text{Area} = \frac{1}{2} \times 9 \times 12 = \frac{1}{2} \times 108 = 54 \text{ cm}^2 \]

4. **Calculate the area of the triangle using the medians**:
Now, we apply the formula to find the area of the original triangle:
\[ A = \frac{4}{3} \times 54 \]

5. **Final calculation**:
\[ A = \frac{4 \times 54}{3} = \frac{216}{3} = 72 \text{ cm}^2 \]

Thus, the area of the triangle is **72 cm²**.