A right triangle has hypotenuse `x` cm and one side of length `y` cm . If `x-y=1 cm`, find the length of the third side of the triangle.

Answer

To solve the problem step by step, we will use the information given about the right triangle and apply the Pythagorean theorem.

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let the hypotenuse be \( x \) cm.
- Let one side be \( y \) cm.
- The other side, which we need to find, will be denoted as \( z \) cm.

2. **Use the Relationship Given**:
- We are given that \( x - y = 1 \) cm.
- From this, we can express \( x \) in terms of \( y \):
\[ x = y + 1 \]

3. **Apply the Pythagorean Theorem**:
- According to the Pythagorean theorem:
\[ x^2 = y^2 + z^2 \]
- Substitute \( x \) with \( y + 1 \):
\[ (y + 1)^2 = y^2 + z^2 \]

4. **Expand the Left Side**:
- Expanding \( (y + 1)^2 \):
\[ y^2 + 2y + 1 = y^2 + z^2 \]

5. **Simplify the Equation**:
- Now, subtract \( y^2 \) from both sides:
\[ 2y + 1 = z^2 \]

6. **Solve for \( z \)**:
- Rearranging gives:
\[ z^2 = 2y + 1 \]
- Taking the square root of both sides:
\[ z = \sqrt{2y + 1} \]

### Final Answer:
The length of the third side \( z \) of the triangle is:
\[ z = \sqrt{2y + 1} \text{ cm} \]