Hard geometry problem, please show and explain all work

Question

The perimeter of a right triangle with side lengths that are integers, and having the same area as a rectangle with dimensions 36 cm by 45 cm is, in cm

Answer 1

Let the sides of the triangle be a, b, and c, with c as the hypotenuse.  So we need area of rectangle (36x45) to be equal to the area of the triangle a*b/2.  And we need a² + b² = c², with all three as integers.  It's this second part, specifically the fact that c must be an integer, that narrows down the possibilities.  So,

 

ab/2 = 36*45

ab = 3240 = 2³*3⁴*5

 

 

There are 20 unique solutions to a*b=3240, so testing possibilities is an option.  I'd prefer to do number theory, but the only obvious insight I get from number theory is that one of a or b must be divisible by 8.  So that limits our possibilities to 10.  Testing those in excel, I get that 120*27 = 3240 and 120² + 27² = 123² as the only solution.  Therefore, the perimeter is 120+27+123 = 270.

Answer 2

Area_retangle = 1620 cm²     (35 times 45)

Area_triangle = (0.5)bh

1620 = (0.5)bh

3240 = bh

 

 More than one pair of integers satisfy.