The hypotenuse of a right triangle is 13 cm and the difference between the other two sides is 7cm. Find the two unknown side of the triangle.

Answer

To solve the problem, we need to find the lengths of the two sides of a right triangle given the hypotenuse and the difference between the two sides.

### Step-by-Step Solution:

1. **Let the two unknown sides be \( a \) and \( b \)**:
- We know that \( a \) is the longer side and \( b \) is the shorter side.
- According to the problem, the difference between the two sides is given as:
\[ a - b = 7 \quad \text{(1)} \]

2. **Use the Pythagorean theorem**:
- The Pythagorean theorem states that:
\[ a^2 + b^2 = h^2 \]
- Here, the hypotenuse \( h = 13 \) cm, so:
\[ a^2 + b^2 = 13^2 = 169 \quad \text{(2)} \]

3. **Express \( a \) in terms of \( b \)**:
- From equation (1), we can express \( a \) as:
\[ a = b + 7 \quad \text{(3)} \]

4. **Substitute equation (3) into equation (2)**:
- Replace \( a \) in equation (2) with the expression from equation (3):
\[ (b + 7)^2 + b^2 = 169 \]

5. **Expand and simplify**:
- Expanding \( (b + 7)^2 \):
\[ b^2 + 14b + 49 + b^2 = 169 \]
- Combine like terms:
\[ 2b^2 + 14b + 49 = 169 \]
- Subtract 169 from both sides:
\[ 2b^2 + 14b + 49 - 169 = 0 \]
\[ 2b^2 + 14b - 120 = 0 \]

6. **Divide the entire equation by 2**:
- To simplify, divide everything by 2:
\[ b^2 + 7b - 60 = 0 \quad \text{(4)} \]

7. **Factor the quadratic equation (4)**:
- We need two numbers that multiply to \(-60\) and add to \(7\). These numbers are \(12\) and \(-5\):
\[ (b + 12)(b - 5) = 0 \]

8. **Solve for \( b \)**:
- Setting each factor to zero gives:
\[ b + 12 = 0 \quad \Rightarrow \quad b = -12 \quad \text{(not valid, as side lengths cannot be negative)} \]
\[ b - 5 = 0 \quad \Rightarrow \quad b = 5 \]

9. **Find \( a \) using equation (3)**:
- Substitute \( b = 5 \) back into equation (3):
\[ a = b + 7 = 5 + 7 = 12 \]

10. **Conclusion**:
- The lengths of the two sides of the triangle are:
\[ a = 12 \, \text{cm}, \quad b = 5 \, \text{cm} \]

### Final Answer:
The two unknown sides of the triangle are \( 12 \, \text{cm} \) and \( 5 \, \text{cm} \).