Respuesta :
Let's call the legs [tex] l_1 [/tex] and [tex] l_2 [/tex], [tex] l_1 [/tex] being the shortest, and the hypotenuse [tex] h [/tex].
The first sentence translates to
[tex] l_2 = 5l_1+19 [/tex]
The second sentence translates to
[tex] h = 5l_1+20 [/tex]
So, we can express all sides in terms of [tex] l_1 [/tex] and substitute the expressions in the Pythagorean threorem:
[tex] h^2 = l_1^2+l_2^2 \to (5l_1+20)^2 = l_1^2 + (5l_1+19)^2 [/tex]
Expand the squares:
[tex] 25 l_1^2 + 200 l_1 + 400 = l_1^2 + 25 l_1^2 + 190 l_1 + 361 [/tex]
Bring everything to the left hand side and simplify:
[tex] l_1^2 - 10 l_1 - 39 = 0 [/tex]
This equation has solutions [tex] l_1 = -3 [/tex] and [tex] l_1 = 13 [/tex].
We can't accept negative lengths, so the answer is [tex] l_1 = 13 [/tex].
From here, it's easy to get the other sides:
[tex] l_2 = 5l_1+19 = 5 \cdot 13 + 19 = 84 [/tex]
[tex] h = 5l_1+20 = 5 \cdot 13 + 20 = 85 [/tex]
