Question 1199461
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(Corrected response -- typo in original response)....<br>
Find the answer using Heron's formula for the area of a triangle in terms of the lengths of the three sides.<br>
{{{A=sqrt((s)(s-a)(s-b)(s-c))}}}<br>
s is the semi-perimeter; a, b, and c are the three side lengths.<br>
In this problem, the perimeter is 54 and one side length is 17.  So<br>
let x = second side length
then 54-(17+x) = 37-x = third side length<br>
The semi-perimeter s is 27; the area is<br>
{{{A=sqrt((27)(27-17)(27-x)(27-(37-x)))}}}
{{{A=sqrt((27)(10)(27-x)(x-10))}}}<br>
The area is an integer; and all three side lengths are integers.<br>
The expression for the area shows that x is greater than 10 and less than 27; trying different integer values for x (either manually or using some kind of calculator) shows that the other two side lengths are {{{cross(24)}}} 25 and {{{cross(13)}}} 12.<br>
ANSWER: the length of the longest side is {{{cross(24)}}} 25.<br>