Find the area of `Delta` whose length of medians is 13 cm, 14 cm , 15 cm.

Answer

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

\[ \text{Area} = \frac{4}{3} \sqrt{s(s-a)(s-b)(s-c)} \]

where \( s \) is the semi-perimeter of the triangle formed by the medians \( a \), \( b \), and \( c \).

### Step-by-Step Solution:

1. **Identify the lengths of the medians**:
- Let \( a = 13 \) cm, \( b = 14 \) cm, and \( c = 15 \) cm.

2. **Calculate the semi-perimeter \( s \)**:
\[ s = \frac{a + b + c}{2} = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 \text{ cm} \]

3. **Substitute the values into the area formula**:
\[ \text{Area} = \frac{4}{3} \sqrt{s(s-a)(s-b)(s-c)} \]

4. **Calculate \( s-a \), \( s-b \), and \( s-c \)**:
- \( s - a = 21 - 13 = 8 \)
- \( s - b = 21 - 14 = 7 \)
- \( s - c = 21 - 15 = 6 \)

5. **Substitute these values into the area formula**:
\[ \text{Area} = \frac{4}{3} \sqrt{21 \times 8 \times 7 \times 6} \]

6. **Calculate the product inside the square root**:
\[ 21 \times 8 = 168 \]
\[ 168 \times 7 = 1176 \]
\[ 1176 \times 6 = 7056 \]

7. **Now calculate the square root**:
\[ \sqrt{7056} \]
- To simplify \( 7056 \), we can factor it:
\[ 7056 = 2^4 \times 3^2 \times 7^2 \]
- Therefore, \( \sqrt{7056} = 2^2 \times 3 \times 7 = 4 \times 3 \times 7 = 84 \).

8. **Substitute back into the area formula**:
\[ \text{Area} = \frac{4}{3} \times 84 = \frac{336}{3} = 112 \text{ cm}^2 \]

### Final Answer:
The area of the triangle is \( 112 \text{ cm}^2 \).

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