Find the area of `Delta` whose length of median are 3,4,5

Answer

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

### Step 1: Identify the lengths of the medians
Let the lengths of the medians be:
- \( m_a = 3 \)
- \( m_b = 4 \)
- \( m_c = 5 \)

### Step 2: Use the formula for the area of a triangle in terms of its medians
The area \( A \) of a triangle whose medians are \( m_a, m_b, m_c \) can be calculated using the formula:
\[ A = \frac{4}{3} \sqrt{s_m (s_m - m_a)(s_m - m_b)(s_m - m_c)} \]
where \( s_m = \frac{m_a + m_b + m_c}{2} \).

### Step 3: Calculate \( s_m \)
First, calculate \( s_m \):
\[ s_m = \frac{3 + 4 + 5}{2} = \frac{12}{2} = 6 \]

### Step 4: Calculate the area using the formula
Now substitute the values into the area formula:
\[ A = \frac{4}{3} \sqrt{6(6 - 3)(6 - 4)(6 - 5)} \]
Calculating the terms inside the square root:
\[ A = \frac{4}{3} \sqrt{6 \times 3 \times 2 \times 1} \]
\[ A = \frac{4}{3} \sqrt{36} \]
\[ A = \frac{4}{3} \times 6 \]
\[ A = \frac{24}{3} = 8 \]

### Conclusion
The area of the triangle is \( 8 \) square units.

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