The length of the three sides of a right angled triangle are `(x-2) cm, x cm and (x + 2) cm `respectively. Then the value of x is

Options

A. 10
B. 8
C. 4
D. 0

Answer

To solve the problem, we need to find the value of \( x \) for the sides of a right-angled triangle given as \( (x-2) \) cm, \( x \) cm, and \( (x+2) \) cm.

### Step-by-Step Solution:

1. **Identify the Hypotenuse**:
In a right-angled triangle, the longest side is the hypotenuse. Here, the sides are \( (x-2) \), \( x \), and \( (x+2) \). The hypotenuse will be \( (x+2) \) since it is the largest.

2. **Apply the Pythagorean Theorem**:
According to the Pythagorean theorem, we have:
\[ \text{(Hypotenuse)}^2 = \text{(Side 1)}^2 + \text{(Side 2)}^2 \]
Substituting the sides, we get:
\[ (x + 2)^2 = (x - 2)^2 + x^2 \]

3. **Expand the Squares**:
Now, we will expand both sides of the equation:
\[ (x + 2)^2 = x^2 + 4x + 4 \]
\[ (x - 2)^2 = x^2 - 4x + 4 \]
Thus, the equation becomes:
\[ x^2 + 4x + 4 = (x^2 - 4x + 4) + x^2 \]

4. **Combine Like Terms**:
On the right side, we combine the terms:
\[ x^2 + 4x + 4 = 2x^2 - 4x + 4 \]

5. **Rearrange the Equation**:
To solve for \( x \), we will move all terms to one side:
\[ x^2 + 4x + 4 - 2x^2 + 4x - 4 = 0 \]
Simplifying this gives:
\[ -x^2 + 8x = 0 \]

6. **Factor the Equation**:
Factoring out \( x \):
\[ x(-x + 8) = 0 \]

7. **Solve for \( x \)**:
Setting each factor to zero gives:
\[ x = 0 \quad \text{or} \quad -x + 8 = 0 \implies x = 8 \]
Since \( x \) must be positive (as it represents a length), we take \( x = 8 \).

### Final Answer:
The value of \( x \) is \( 8 \).