Smallest Side of a Right Triangle

The sides of a right triangle are all integers. Two of them are odd numbers that differ       by 50. What is the smallest possible value for the third side?
Source: NCTM Mathematics Teacher, February 2006

SOLUTION
We look at primitive triples of a right triangle because they are integers that have the smallest possible values. A property of primitive triples states that the length of one leg is odd, that of the other leg is even and that of the hypotenuse is odd.
Let a represent the odd leg, b the even leg, and c the odd hypotenuse. By the Pythagorean theorem
a^2+b^2=c^2
a^2+b^2=(a+50)^2
b^2=(a+50)^2-a^2
=(a+50+a)(a+50-a)
=(2a+50)(50)
=(a+25)100
=(a+25)10^2
a+25 must be a perfect square. The smallest perfect square greater than 25 is 36
a+25=36
a=11
11^2+b^2=61^2
b^2=61^2-11^2
=3600
b=60
The smallest possible value for the third side equals 60.

Answer: 60

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About mvtrinh

Retired high school math teacher.
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