The sides of a right angled triangle are equal to three consecutive numbers expressed in centimeters. What can be the area of such a triangle?

Options

A. `6 cm^(2)`
B. `8 cm^(2)`
C. `10 cm^(2)`
D. `12 cm^(2)`

Answer

To solve the problem of finding the area of a right-angled triangle with sides equal to three consecutive integers, we can follow these steps:

### Step 1: Define the sides of the triangle
Let the sides of the triangle be represented as:
- One side (base) = \( x \)
- Second side (height) = \( x + 1 \)
- Hypotenuse = \( x + 2 \)

### Step 2: Apply the Pythagorean theorem
Since this is a right-angled triangle, we can apply the Pythagorean theorem:
\[ \text{(Hypotenuse)}^2 = \text{(Base)}^2 + \text{(Height)}^2 \]
Substituting the sides into the equation gives:
\[ (x + 2)^2 = x^2 + (x + 1)^2 \]

### Step 3: Expand the equation
Now, we expand both sides:
\[ (x + 2)^2 = x^2 + 4x + 4 \]
\[ (x + 1)^2 = x^2 + 2x + 1 \]
Thus, the equation becomes:
\[ x^2 + 4x + 4 = x^2 + x^2 + 2x + 1 \]
This simplifies to:
\[ x^2 + 4x + 4 = 2x^2 + 2x + 1 \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ 0 = 2x^2 + 2x + 1 - x^2 - 4x - 4 \]
\[ 0 = x^2 - 2x - 3 \]

### Step 5: Factor the quadratic equation
Now we factor the quadratic:
\[ x^2 - 2x - 3 = (x - 3)(x + 1) = 0 \]
Setting each factor to zero gives:
\[ x - 3 = 0 \quad \text{or} \quad x + 1 = 0 \]
Thus, \( x = 3 \) or \( x = -1 \). Since side lengths cannot be negative, we take \( x = 3 \).

### Step 6: Calculate the sides of the triangle
Now we can find the lengths of the sides:
- Base (BC) = \( x + 1 = 3 + 1 = 4 \) cm
- Height (AB) = \( x = 3 \) cm
- Hypotenuse (AC) = \( x + 2 = 3 + 2 = 5 \) cm

### Step 7: Calculate the area of the triangle
The area \( A \) of a right-angled triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{Base} \times \text{Height} \]
Substituting the values:
\[ A = \frac{1}{2} \times 4 \times 3 = \frac{12}{2} = 6 \text{ cm}^2 \]

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

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