The sides of a right-angled triangle are in the ratio x : ( x + 1 ) : ( x + 2 ) . What is the perimeter of the triangle?

Options

A. `10 cm`
B. `12 cm`
C. `15 cm`
D. none of the above

Answer

To solve the problem of finding the perimeter of a right-angled triangle with sides in the ratio \( x : (x + 1) : (x + 2) \), we can follow these steps:

### Step 1: Identify the sides of the triangle
The sides of the triangle can be represented as:
- Side 1: \( x \)
- Side 2: \( x + 1 \)
- Side 3 (hypotenuse): \( x + 2 \)

### Step 2: Apply the Pythagorean theorem
In a right-angled triangle, the Pythagorean theorem states that:
\[ \text{(Hypotenuse)}^2 = \text{(Base)}^2 + \text{(Height)}^2 \]
Here, we can assume:
\[ (x + 2)^2 = x^2 + (x + 1)^2 \]

### Step 3: Expand the equation
Now, let's expand both sides:
- Left side:
\[ (x + 2)^2 = x^2 + 4x + 4 \]
- Right side:
\[ x^2 + (x + 1)^2 = x^2 + (x^2 + 2x + 1) = 2x^2 + 2x + 1 \]

### Step 4: Set up the equation
Now we can set the two sides equal to each other:
\[ x^2 + 4x + 4 = 2x^2 + 2x + 1 \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ 0 = 2x^2 + 2x + 1 - x^2 - 4x - 4 \]
This simplifies to:
\[ 0 = x^2 - 2x - 3 \]

### Step 6: Factor the quadratic equation
Next, we can factor the quadratic equation:
\[ x^2 - 2x - 3 = (x - 3)(x + 1) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor to zero gives us:
\[ x - 3 = 0 \quad \text{or} \quad x + 1 = 0 \]
Thus, \( x = 3 \) or \( x = -1 \). Since \( x \) represents a length, we discard \( x = -1 \).

### Step 8: Find the lengths of the sides
Now substituting \( x = 3 \):
- Side 1: \( x = 3 \)
- Side 2: \( x + 1 = 4 \)
- Side 3 (hypotenuse): \( x + 2 = 5 \)

### Step 9: Calculate the perimeter
The perimeter \( P \) of the triangle is given by the sum of its sides:
\[ P = 3 + 4 + 5 = 12 \]

### Final Answer
The perimeter of the triangle is \( 12 \) units.

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