SOLUTION: If (6-x), (13-x), and (14-x) are the lengths of the sides of a right triangle, find the value of x.

Question 340309: If (6-x), (13-x), and (14-x) are the lengths of the sides of a right triangle, find the value of x.
Answer by J2R2R(94) (Show Source): You can put this solution on YOUR website!
We have three sides 6 - x, 13 - x, 14 - x

14 - x is the largest so this must be the hypotenuse.

Therefore using Pythagoras we have
(6 - x)^2 + (13 - x)^2 = (14 - x)^2
36 - 12x + x^2 + 169 - 26x + x^2 = 196 - 28x + x^2

which gives:
x^2 - 10x + 9 = 0

Factorising gives (x - 1)(x - 9) = 0
x = 1 and x = 9 give solutions to the equation.

With x = 1 we have sides 5, 12 and 13 which is reasonable; but with x = 9 we have sides -3, 4 and 5 which does satisfy the equation but as we are talking about lengths (which are all positive) we disregard this answer.

Therefore the sides are 5, 12 and 13 with x = 1.
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