SOLUTION: what is the lengths of the three sides of a right triangle are consecutive integers. find them can you plz help me and show your work

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Question 582215: what is the lengths of the three sides of a right triangle are consecutive integers. find them
can you plz help me and show your work
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I know that the most classic Pythagorean triple is (3,4,5), but I will pretend that I do not know.
Let the three consecutive integers be x-1, x, and x+1.
If they are lengths of the three sides of a right triangle, the largest, x+1, is the length of the hypotenuse.
Pythagoras theorem says that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In this case,
%28x-1%29%5E2%2Bx%5E2=%28x%2B1%29%5E2 --> x%5E2-2x%2B1%2Bx%5E2=x%5E2%2B2x%2B1 --> 2x%5E2-2x%2B1=x%5E2%2B2x%2B1 --> 2x%5E2-2x%2B1-x%5E2-2x-1=x%5E2%2B2x%2B1-x%5E2-2x-1 --> x%5E2-4x=0 --> x%28x-4%29=0
The solutions to that equation are x=0 and x=4.
x=0 does not make sense for the middle integer, because the length of the shortest side would be -1, and we expect those lengths to be positive.
So the middle integer is highlight%28x=4%29 and the other two numbers are
highlight%28x-1=3%29 and highlight%28x%2B1=5%29.
The lengths of the three sides of the right triangle are the consecutive integers 3, 4, and 5.