An architect is designing a building with a right triangular foot print. the hypotenuse of the triangle is eighty feet longer than one leg of the triangle and forty feet longer than the other leg use the pythagorean theorem to find the dimensions of the footprint of the building

Let the hypotenuse = c
One of the legs = c - 80
The other leg = c - 40

(c - 80)^2 + (c - 40)^2 = c^2 Expand the left hand side.
c^2 - 160c + 6400 + c^2 - 80c + 1600 = c^2 Subtract c^2 from both sides.
c^2 - 160c + 6400 + c^2 - c^2 - 80c + 1600 = 0        Combine like terms.
c^2 - 240c + 8000 = 0
(c - 200)(c  - 40) is how that factors. It looks like you have two solutions. You don't. Not really
(c - 200) = 0
c - 200 = 0
c = 200

The other solution is
c - 40 = 0
c = 40

The last solution does not work. Why? It's because the let that was forty smaller than c would give that leg a length of 40 - 40 = 0

So c = 200

The lengths of the triangle are
c (the hypotenuse) = 200
The next leg = 200 - 40 = 160
The third leg = 200 - 80 = 120

Check
a^2 + b^2 = c^2
120^2 + 160^2 = 200^2
14400 + 25600 = 40000
40000 = 40000
They check