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# Triangle Sides

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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
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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