In a right-angled triangle, the square of the longest side is 625 sq. cm. if the length of the second side is 7 cm, find the length of the third side of the triangle.

Answer

To solve the problem, we will use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
- Let \( AC \) be the hypotenuse (longest side).
- Let \( AB \) be one of the other sides, which is given as \( 7 \) cm.
- Let \( BC \) be the side we need to find.

2. **Write down the given information**:
- The square of the hypotenuse \( AC^2 = 625 \) sq. cm.
- The length of side \( AB = 7 \) cm.

3. **Use the Pythagorean theorem**:
- According to the Pythagorean theorem:
\[ AB^2 + BC^2 = AC^2 \]
- Substitute the known values into the equation:
\[ 7^2 + BC^2 = 625 \]

4. **Calculate \( 7^2 \)**:
- \( 7^2 = 49 \).
- Now the equation becomes:
\[ 49 + BC^2 = 625 \]

5. **Isolate \( BC^2 \)**:
- Subtract \( 49 \) from both sides:
\[ BC^2 = 625 - 49 \]
- Calculate \( 625 - 49 \):
\[ BC^2 = 576 \]

6. **Find \( BC \)**:
- Take the square root of both sides:
\[ BC = \sqrt{576} \]
- Calculate \( \sqrt{576} \):
\[ BC = 24 \text{ cm} \]

7. **Conclusion**:
- The length of the third side \( BC \) is \( 24 \) cm.

### Final Answer:
The length of the third side of the triangle is \( 24 \) cm.

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