We have the sides of a triangle: x, x+1, x+2.
The Pythagorean Theorem tells us that, in a right triangle, the following expression is true:
c^(2) = a^(2) + b^(2)
Let be the sides (legs of the triangle) x and x+1.
Then:
(x+2)^2 = x^(2) + (x+1)^2
We need to expand this:
x^(2) +4x + 4 = x^(2) + x^(2) +2x + 1
Rearrange the previous expression (grouping similar terms):
4x - 2x + 4 - 1 = 2x^(2) - x^(2)
2x + 3 = x^(2)
x^(2) - 2x - 3 = 0 [1]
Which is a quadratic equation. We need to find the values for this equation.
We can solve this equation using the quadratic formula and other methods.
One of the methods is as follows: we need to find two values that if we sum them, we obtain -2. If we multiply them, we obtain -3. Then, 1 -3 =-2 (-2x) and 1*(-3) = -3 (independent term).
The values for x are x = -1 or x = 3.
Since x = -1 is a negative number, it does not have sense. Therefore, the right value for x is 3 (x = 3).
The values for the sides of the right triangle are then:
x = 3, x+1 = 3+1 =4, x+2 = 3+2 =5 or
3, 4, 5.