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. `24sqrt(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:

\[ \text{Area} = \frac{4}{3} \times \sqrt{s_m \times (s_m - m_1) \times (s_m - m_2) \times (s_m - m_3)} \]

where \( m_1, m_2, m_3 \) are the lengths of the medians, and \( s_m \) is the semi-perimeter of the triangle formed by the medians:

\[ s_m = \frac{m_1 + m_2 + m_3}{2} \]

### Step 1: Calculate the semi-perimeter \( s_m \)

Given the lengths of the medians:
- \( m_1 = 18 \) cm
- \( m_2 = 24 \) cm
- \( m_3 = 30 \) cm

Calculate \( s_m \):

\[ s_m = \frac{18 + 24 + 30}{2} = \frac{72}{2} = 36 \text{ cm} \]

### Step 2: Substitute values into the area formula

Now, substitute \( s_m \) and the median lengths into the area formula:

\[ \text{Area} = \frac{4}{3} \times \sqrt{36 \times (36 - 18) \times (36 - 24) \times (36 - 30)} \]

### Step 3: Calculate each term inside the square root

Calculate \( (s_m - m_1) \), \( (s_m - m_2) \), and \( (s_m - m_3) \):

- \( s_m - m_1 = 36 - 18 = 18 \)
- \( s_m - m_2 = 36 - 24 = 12 \)
- \( s_m - m_3 = 36 - 30 = 6 \)

Now substitute these values:

\[ \text{Area} = \frac{4}{3} \times \sqrt{36 \times 18 \times 12 \times 6} \]

### Step 4: Calculate the product inside the square root

Calculate \( 36 \times 18 \times 12 \times 6 \):

1. \( 36 \times 18 = 648 \)
2. \( 12 \times 6 = 72 \)
3. Now multiply \( 648 \times 72 \):

\[ 648 \times 72 = 46656 \]

### Step 5: Take the square root

Now find the square root:

\[ \sqrt{46656} = 216 \]

### Step 6: Calculate the area

Now substitute back into the area formula:

\[ \text{Area} = \frac{4}{3} \times 216 = 288 \text{ cm}^2 \]

Thus, the area of the triangle is:

\[ \boxed{288 \text{ cm}^2} \]