Answer:
I will answer in English.
We can prove that the angle APS is a triangle rectangle.
Remember that for a triangle rectangle of catheti A and B, and hypotenuse H, the Pythagorean's theorem says that:
A^2 + B^2 = H^2
In this case, we can assume that the hypotenuse is the longer side, AS, and the other two sides are the catheti.
Then we have:
H = 5x + 10
A = 3x + 6
B = 4x + 8
Now let's write the equation from the theorem, and let's see if its true.
A^2 + B^2 = H^2
( 3x + 6 )^2 + (4x + 8)^2 = (5x + 10)^2
So we can start with:
( 3x + 6 )^2 + (4x + 8)^2
And try to "transform" this into:
(5x + 10)^2
First, let's expand it:
((3x)^2 + 2*(3x)*6 + 6^2) + ( (4x)^2 + 2*(4x)*8 + 8^2)
9x^2 + 24x + 36 + 16x^2 + 64x + 64
25x^2 + 40x + 100
Now we can complete squares on the left side, by writing:
(5x)^2 + 2*10*(5x) + 10^2
(5x + 10)^2
Then we saw that the equation is true for every value of x, then we just prove that the triangle fulfills the theorem, thus, the triangle is a triangle rectangle.
