## 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: $60$