In `triangleABC, /_B=90^(@)`. Find the sides of the triangle if:
`AB=(x-3) cm, BC=(x+4) cm" and "AC=(x+6) cm`.

Answer

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

### Step-by-Step Solution:

1. **Identify the sides of the triangle:**
- Given:
- \( AB = (x - 3) \) cm
- \( BC = (x + 4) \) cm
- \( AC = (x + 6) \) cm

2. **Apply the Pythagorean theorem:**
Since angle B is 90 degrees, we can set up the equation:
\[ AB^2 + BC^2 = AC^2 \]
Substituting the expressions for the sides:
\[ (x - 3)^2 + (x + 4)^2 = (x + 6)^2 \]

3. **Expand the squares:**
- Expanding \( (x - 3)^2 \):
\[ (x - 3)^2 = x^2 - 6x + 9 \]
- Expanding \( (x + 4)^2 \):
\[ (x + 4)^2 = x^2 + 8x + 16 \]
- Expanding \( (x + 6)^2 \):
\[ (x + 6)^2 = x^2 + 12x + 36 \]

4. **Combine the expanded terms:**
Substitute the expanded forms back into the equation:
\[ (x^2 - 6x + 9) + (x^2 + 8x + 16) = (x^2 + 12x + 36) \]
Combine like terms on the left side:
\[ 2x^2 + 2x + 25 = x^2 + 12x + 36 \]

5. **Rearrange the equation:**
Move all terms to one side to set the equation to zero:
\[ 2x^2 + 2x + 25 - x^2 - 12x - 36 = 0 \]
Simplifying gives:
\[ x^2 - 10x - 11 = 0 \]

6. **Factor the quadratic equation:**
We need to factor \( x^2 - 10x - 11 \):
\[ (x - 11)(x + 1) = 0 \]

7. **Solve for x:**
Setting each factor to zero gives:
\[ x - 11 = 0 \quad \text{or} \quad x + 1 = 0 \]
Thus, \( x = 11 \) or \( x = -1 \). Since side lengths cannot be negative, we take \( x = 11 \).

8. **Calculate the sides of the triangle:**
Substitute \( x = 11 \) back into the expressions for the sides:
- \( AB = 11 - 3 = 8 \) cm
- \( BC = 11 + 4 = 15 \) cm
- \( AC = 11 + 6 = 17 \) cm

### Final Answer:
The sides of triangle ABC are:
- \( AB = 8 \) cm
- \( BC = 15 \) cm
- \( AC = 17 \) cm