If the sides of a right angled triangle are three consecutive integers, then the length of the smallest side is

Options

A. a) 3 units
B. b) 2 units
C. c) 4 units
D. d) 5 units

Answer

To solve the problem of finding the length of the smallest side of a right-angled triangle with sides as 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 three consecutive integers. We can denote the smallest side as \( x \). Therefore, the sides of the triangle can be represented as:
- Smallest side: \( x \)
- Middle side: \( x + 1 \)
- Largest side (hypotenuse): \( x + 2 \)

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

### Step 3: Expand the equation
Now, we will expand both sides of the equation:
\[ (x + 2)^2 = x^2 + (x + 1)^2 \]
Expanding the left side:
\[ x^2 + 4x + 4 \]
Expanding the right side:
\[ x^2 + (x^2 + 2x + 1) = 2x^2 + 2x + 1 \]
So, we have:
\[ x^2 + 4x + 4 = 2x^2 + 2x + 1 \]

### Step 4: Rearrange the equation
Now, we will rearrange the equation to bring all terms to one side:
\[ x^2 + 4x + 4 - 2x^2 - 2x - 1 = 0 \]
This simplifies to:
\[ -x^2 + 2x + 3 = 0 \]
Multiplying through by -1 gives:
\[ x^2 - 2x - 3 = 0 \]

### Step 5: Factor the quadratic equation
Next, we will factor the quadratic equation:
\[ x^2 - 2x - 3 = (x - 3)(x + 1) = 0 \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives:
1. \( x - 3 = 0 \) → \( x = 3 \)
2. \( x + 1 = 0 \) → \( x = -1 \) (not a valid solution since side lengths cannot be negative)

Thus, the only valid solution is:
\[ x = 3 \]

### Conclusion
The length of the smallest side of the triangle is \( 3 \).

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