# If x, x + 2 and x + 1 are three sides of a right triangle, what is x

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### 2 Answers

The sides of a right triangle are x, x + 1 and x + 2. Using Pythagoras' Theorem, (x + 2)^2 = (x + 1)^2 + x^2

=> x^2 + 4x + 4 = x^2 + 2x + 1 + x^2

=> x^2 - 2x - 3 = 0

=> x^2 - 3x + x - 3 = 0

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

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

=> x = -1 and x = 3

The length of a side cannot be negative, ignore x = -1.

**The length of the sides of the right triangle are 3, 4 and 5**

The sides of a right triangle are x, x + 2 and x + 1. Of the three terms the largest term is x + 2. This is the hypotenuse of the right triangle. Using Pythagoras theorem, the sum of the squares of the sides other than the hypotenuse is equal to the square of the hypotenuse.

Here, this gives:

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

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

x^2 - 2x - 3 = 0

x^2 - 3x + x - 3 = 0

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

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

Ignore the negative root as the sides of a triangle cannot have negative length. This gives x = 3

The sides of the triangle are 3, 4 and 5