The side (in cm) of a right triangle are x - 1, x and x + 1. The area of triangle is

Options

A. `5 cm^2`
B. `3 cm^2`
C. `6 cm^2`
D. None of these

Answer

To find the area of the right triangle with sides \(x - 1\), \(x\), and \(x + 1\), we can follow these steps:

### Step 1: Identify the sides of the triangle
Given the sides are \(x - 1\), \(x\), and \(x + 1\), we can identify that:
- \(x - 1\) is one leg,
- \(x\) is the other leg,
- \(x + 1\) is the hypotenuse (since it is the largest side).

### Step 2: Apply the Pythagorean theorem
According to the Pythagorean theorem, for a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Therefore, we can write:
\[ (x - 1)^2 + x^2 = (x + 1)^2 \]

### Step 3: Expand the equation
Now, expand both sides:
\[ (x - 1)^2 = x^2 - 2x + 1 \]
\[ x^2 = x^2 \]
\[ (x + 1)^2 = x^2 + 2x + 1 \]
Substituting these into the equation gives:
\[ (x^2 - 2x + 1) + x^2 = (x^2 + 2x + 1) \]

### Step 4: Simplify the equation
Combine like terms:
\[ 2x^2 - 2x + 1 = x^2 + 2x + 1 \]
Now, subtract \(x^2 + 2x + 1\) from both sides:
\[ 2x^2 - 2x + 1 - x^2 - 2x - 1 = 0 \]
This simplifies to:
\[ x^2 - 4x = 0 \]

### Step 5: Factor the equation
Factor out \(x\):
\[ x(x - 4) = 0 \]
This gives us two solutions:
\[ x = 0 \quad \text{or} \quad x = 4 \]

### Step 6: Determine valid solutions
Since the sides of a triangle cannot be zero or negative, we discard \(x = 0\). Thus, we have:
\[ x = 4 \]

### Step 7: Calculate the sides of the triangle
Substituting \(x = 4\) back into the expressions for the sides:
- First side: \(x - 1 = 4 - 1 = 3\)
- Second side: \(x = 4\)
- Third side: \(x + 1 = 4 + 1 = 5\)

### Step 8: Calculate the area of the triangle
The area \(A\) of a right triangle is given by:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take the base as \(3\) and the height as \(4\):
\[ A = \frac{1}{2} \times 3 \times 4 = 6 \text{ cm}^2 \]

### Final Answer
The area of the triangle is \(6 \text{ cm}^2\).

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