In a right angled triangle ABC, right angled at B, AB=3, BC=x+2 and AC=x+3. Then find the value of x:

Options

A. 4
B. 2
C. 1
D. 3

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 side opposite the right angle) 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:**
- In triangle ABC, we have:
- AB = 3 (one leg)
- BC = x + 2 (the other leg)
- AC = x + 3 (the hypotenuse)

2. **Apply the Pythagorean theorem:**
- According to the theorem, we can write:
\[ AC^2 = AB^2 + BC^2 \]
- Substituting the values:
\[ (x + 3)^2 = 3^2 + (x + 2)^2 \]

3. **Expand both sides of the equation:**
- Left side:
\[ (x + 3)^2 = x^2 + 6x + 9 \]
- Right side:
\[ 3^2 = 9 \quad \text{and} \quad (x + 2)^2 = x^2 + 4x + 4 \]
- Therefore, the right side becomes:
\[ 9 + (x^2 + 4x + 4) = x^2 + 4x + 13 \]

4. **Set the expanded equations equal to each other:**
\[ x^2 + 6x + 9 = x^2 + 4x + 13 \]

5. **Simplify the equation:**
- Cancel \(x^2\) from both sides:
\[ 6x + 9 = 4x + 13 \]
- Rearranging gives:
\[ 6x - 4x = 13 - 9 \]
- This simplifies to:
\[ 2x = 4 \]

6. **Solve for x:**
- Divide both sides by 2:
\[ x = 2 \]

### Final Answer:
The value of \(x\) is \(2\).