One leg of a right triangle is 4 less than the other leg . the square of the hypotenuse for the right triangle is 80 . how long are the legs of the right triangle

To solve this question, let's first make use of the variables in Pythagorean theorem, so that it would be easier to use, when using the formula.

a = x
b = x - 4
c^2 = 80

First find the square root of 80, to obtain the length of the hypotenuse, in this case it would be 8.944. Now we can use Pythagorean theorem to solve for x.

a^2 + b^2 = c^2
(x)^2 + (x-4)^2 = 80
x^2 + x^2 - 8x + 16 = 80
2x^2 - 8x + 16 = 80
2x^2 - 8x = 64
2x^2 - 8x - 64 = 0
2(x^2 - 4x - 32) = 0
2(x-8)(x+4) = 0

The solutions for x would be 8 and -4 and since the length cannot be negative we only have one value for x and that is 8.

Thus
a = x = 8
b = x - 4 = 8 - 4 = 4
a = 8
b = 4.

These would be the lengths for right angle triangle.