We are given that the sides of a right triangle are $x$, $x+1$, and $x+2$. We need to find the side lengths of the triangle by solving for $x$.

1. Problem Description

We are given that the sides of a right triangle are $x$, $x+1$, and $x+2$. We need to find the side lengths of the triangle by solving for $x$.

2. Solution Steps

Since the sides of the triangle are $x$, $x+1$, and $x+2$, and it is a right triangle, we can apply the Pythagorean theorem. Since $x+2$ is the largest side, it is the hypotenuse. Thus, we have

$$x^2 + (x+1)^2 = (x+2)^2$$

Expanding the terms, we get

$$x^2 + (x^2 + 2x + 1) = x^2 + 4x + 4$$

Combining like terms, we have

$$2x^2 + 2x + 1 = x^2 + 4x + 4$$

Subtracting $x^2 + 4x + 4$ from both sides, we have

$$x^2 - 2x - 3 = 0$$

Now, we factor the quadratic equation:

$$(x-3)(x+1) = 0$$

The solutions are $x=3$ and $x=-1$. Since $x$ represents a side length, it must be positive. Therefore, $x=3$.

The side lengths are $x=3$, $x+1=3+1=4$, and $x+2=3+2=5$.

We check if the Pythagorean theorem holds: $3^2 + 4^2 = 9 + 16 = 25 = 5^2$. So the side lengths are 3, 4, and

5.

3. Final Answer

The side lengths of the triangle are 3, 4, and 5.