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A right triangle with integer leg lengths is called cool if the number of square units in its area is equal to three 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?

 Jul 13, 2023
 #1
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I figured it out, the answer is 471 for anyone who is wondering. The legs are x and y, so you get xy/2 = 3x + 3y. xy = 6x + 6y, so xy - 6x - 6y = 0. 

(x - 6)(y - 6) - 36 = 0, so (x - 6)(y - 6) = 36. Then, just calculate all possible values of x and y to find the sum of all areas as 471

 Jul 13, 2023
 #2
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(29, 20, 21) , (30, 18, 24) , (34, 16, 30) , (39, 15, 36) , (50, 14, 48) , (85, 13, 84)> Total = 6 such triangles

 

[1/2 * 20 *21] + [1/2 * 18 * 24] + [1/2 * 16 * 30] + [1/2 * 15 * 36] + [1/2 * 14 * 48] + [1/2 * 13 * 84]==1,818

 Jul 13, 2023
 #3
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HumenBeing is back to cheating! Business as usual.

 Jul 13, 2023

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