The lengths of the sides of a triangle are positive integers. One side has length 17cm and the perimeter of the triangle is 50cm. If the area is also an integer, then the length of the shortest side is.

Hans007
May 7, 2017

#1**+1 **

The lengths of the sides of a triangle are positive integers. One side has length 17cm and the perimeter of the triangle is 50cm. If the area is also an integer, then the length of the shortest side is.

Mmmmm......this one might be tough to figure out.....

This is a "guess and check" answer

Let side a = 17

Call the unknown sides b and c

We know that 17 + b + c = 50 → b + c = 33

Let the other two sides be b = 17 and c = 16

The semi-perimeter = perimeter/ 2 = 50/2 = 25 = s

And by Heron's formula....we have...

Area = sqrt [ s (s - a) (s - b) (s - c) ]

Area = sqrt [ 25 * (25 - 17) (25- 17) (25 - 16)] =

sqrt [ 25 * 8^2 * 9 ] =

sqrt [ 25 * 64 * 9] =

5 * 8 * 3 = 120 cm^2

So...... the two sides a and b are equal = 17 cm each

And the remaining shortest side = c = 16 cm

P.S. - there may be other possibilities ??

CPhill
May 7, 2017