The medians of a triangle ABC are 9 cm, 12 cm and 15 cm respectively. Then the area of the triangle is.

Answer

To find the area of triangle ABC given the lengths of its medians (9 cm, 12 cm, and 15 cm), we can use the formula for the area of a triangle based on its medians.

### Step-by-Step Solution:

1. **Identify the lengths of the medians**:
Let the lengths of the medians be:
- \( m_1 = 9 \) cm
- \( m_2 = 12 \) cm
- \( m_3 = 15 \) cm

2. **Calculate the semi-perimeter (s)**:
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{m_1 + m_2 + m_3}{2} \]
Substituting the values:
\[ s = \frac{9 + 12 + 15}{2} = \frac{36}{2} = 18 \text{ cm} \]

3. **Use the area formula for the triangle based on its medians**:
The area \( A \) of the triangle can be calculated using the formula:
\[ A = \frac{4}{3} \sqrt{s(s - m_1)(s - m_2)(s - m_3)} \]
Substitute the values of \( s \), \( m_1 \), \( m_2 \), and \( m_3 \):
\[ A = \frac{4}{3} \sqrt{18(18 - 9)(18 - 12)(18 - 15)} \]
Simplifying the terms inside the square root:
\[ A = \frac{4}{3} \sqrt{18 \times 9 \times 6 \times 3} \]

4. **Calculate the product under the square root**:
First, calculate \( 9 \times 6 \times 3 \):
\[ 9 \times 6 = 54 \]
\[ 54 \times 3 = 162 \]
Now, multiply by 18:
\[ 18 \times 162 = 2916 \]

5. **Calculate the square root**:
Now, find the square root of 2916:
\[ \sqrt{2916} = 54 \]

6. **Final calculation of the area**:
Substitute back into the area formula:
\[ A = \frac{4}{3} \times 54 = 72 \text{ cm}^2 \]

### Final Answer:
The area of triangle ABC is \( 72 \text{ cm}^2 \).