In a right-angled triangle ABC, right angled at B, `AB = (x)/(2),` BC = x + 2 and AC = x + 3. The value of x is :

Options

A. 5
B. 10
C. 12
D. 14

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 (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:**
- \( AB = \frac{x}{2} \)
- \( BC = x + 2 \)
- \( AC = x + 3 \)

2. **Apply the Pythagorean theorem:**
According to the theorem:
\[ AB^2 + BC^2 = AC^2 \]
Substituting the values of the sides:
\[ \left(\frac{x}{2}\right)^2 + (x + 2)^2 = (x + 3)^2 \]

3. **Expand each term:**
- \( \left(\frac{x}{2}\right)^2 = \frac{x^2}{4} \)
- \( (x + 2)^2 = x^2 + 4x + 4 \)
- \( (x + 3)^2 = x^2 + 6x + 9 \)

Now substituting these expansions back into the equation:
\[ \frac{x^2}{4} + (x^2 + 4x + 4) = (x^2 + 6x + 9) \]

4. **Combine like terms:**
The equation becomes:
\[ \frac{x^2}{4} + x^2 + 4x + 4 = x^2 + 6x + 9 \]

To eliminate the fraction, multiply the entire equation by 4:
\[ x^2 + 4x^2 + 16x + 16 = 4x^2 + 24x + 36 \]

Simplifying gives:
\[ 5x^2 + 16x + 16 = 4x^2 + 24x + 36 \]

5. **Rearranging the equation:**
Move all terms to one side:
\[ 5x^2 - 4x^2 + 16x - 24x + 16 - 36 = 0 \]
This simplifies to:
\[ x^2 - 8x - 20 = 0 \]

6. **Factor the quadratic equation:**
We can factor this equation:
\[ (x - 10)(x + 2) = 0 \]

7. **Find the values of x:**
Setting each factor to zero gives:
\[ x - 10 = 0 \quad \Rightarrow \quad x = 10 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

8. **Determine the valid solution:**
Since \( x \) must be positive (as it represents a length), we discard \( x = -2 \). Thus, the valid solution is:
\[ x = 10 \]

### Final Answer:
The value of \( x \) is \( 10 \).