A right triangle with integer leg lengths is called cool if the number of square units in its area is equal to ten times the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles?

parmen Sep 17, 2024

#1**0 **

*A right triangle with integer leg lengths is called cool if the number of square units in its area is equal to ten times the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles?*

Plotting on Desmos, xy/2 = 10(x+y) is a hyperbola.

I checked the curve for points where x and y are integers.

In the positive quadrant, I found (25,100), (30,60), and (40,40).

This makes the areas 1250, 900, and 800, respectively. **Total = 2950**

I'm considering the (25,100) and (30,60) triangles the same as

the (100,25) and (60,30) triangles, and counted them only once.

_{.}

Bosco Sep 17, 2024