If medians of a triangle have lengths 18 cm, 24 cm and 30 cm, what is the area ( in `cm^(2)` ) of the triangle ?

Options

A. `24 sqrt(6)`
B. 244
C. 288
D. 360

Answer

To find the area of a triangle given the lengths of its medians, we can use the formula:

\[ \text{Area} = \frac{4}{3} \sqrt{s \cdot (s - A) \cdot (s - B) \cdot (s - C)} \]

where \( A, B, C \) are the lengths of the medians, and \( s \) is the semi-perimeter defined as:

\[ s = \frac{A + B + C}{2} \]

### Step-by-Step Solution:

1. **Identify the lengths of the medians:**
- Let \( A = 18 \, \text{cm} \)
- Let \( B = 24 \, \text{cm} \)
- Let \( C = 30 \, \text{cm} \)

2. **Calculate the semi-perimeter \( s \):**
\[ s = \frac{A + B + C}{2} = \frac{18 + 24 + 30}{2} = \frac{72}{2} = 36 \, \text{cm} \]

3. **Substitute the values into the area formula:**
\[ \text{Area} = \frac{4}{3} \sqrt{s \cdot (s - A) \cdot (s - B) \cdot (s - C)} \]

4. **Calculate \( s - A, s - B, s - C \):**
- \( s - A = 36 - 18 = 18 \)
- \( s - B = 36 - 24 = 12 \)
- \( s - C = 36 - 30 = 6 \)

5. **Substitute these values into the area formula:**
\[ \text{Area} = \frac{4}{3} \sqrt{36 \cdot 18 \cdot 12 \cdot 6} \]

6. **Calculate the product inside the square root:**
- First calculate \( 36 \cdot 18 = 648 \)
- Then calculate \( 12 \cdot 6 = 72 \)
- Now multiply \( 648 \cdot 72 \):
\[ 648 \cdot 72 = 46656 \]

7. **Take the square root:**
\[ \sqrt{46656} = 216 \]

8. **Calculate the area:**
\[ \text{Area} = \frac{4}{3} \cdot 216 = 288 \, \text{cm}^2 \]

### Final Answer:
The area of the triangle is \( 288 \, \text{cm}^2 \).