The length of the sides of a triangle are integers and its area is also integer one side is 21 and the perimeter is 48 find the shortest side

Answer 1

the perimeter is 48 and one side is 21

therefore the other two sides add up to 27

use 10 and 17 for the other two sides

the three sides are 10, 17, and 21

use Heron's formula

A=√S(S-A)(S-B)(S-C) which must be an integer (S=(A+B+C)/2)

S=48/2=24

A=√(24)(24-21)(24-10)(24-17)

A=√(24*3*14*7)

A=√7056

A=84 which is an integer, so here is one solution

the shortest side is 10

there may be more solutions

Answer 2

perimeter = 48; one side is 21, so the total of two sides must be 48-21 = 27.

 

One solution would be to divide the 27 into 10 and 17.  But is that the only possible solution??

 

Something to think about.