Find the specific sides of a parallelogram

Question

Two bisectors are drawn from the corners (next to the longest side) of the parallelogram. Both sides of the parallelogram are given. Could you please tell me the steps of calculating the parts on the opposite side of the parallelogram that are cut off by the bisectors? It's easier to understand if you view the picture (solve for x, y and z. a and b are given, the angles are not).

Answer 1

Extend a triangle like this:

If we start at the red corner and move towards the $\alpha$ angle, then the line parallel to $b$ changes length from $0$ to $b$.

The total length of the extension is $b$ because $\alpha+\beta=90^\circ$, due to $2\alpha+2\beta=180^\circ$, and so the reflection is over an angle of $90^\circ$.

At the point we want, $x=b-a$.

Answer 2

Since the angles don't affect $x$, $y$, and $z$ you can just assume that they are 90°. In a rectangle you can easily see that $x = z = a$ and $y = b - 2a$.

Answer 3

AlejandroBergasaAlonso Helped me to notice that there is an isosceles triangle forming with the side and the bisector. That means that x + y = a. The same goes for the other side: y + z = a (both sides are equal singe it's a parallelogram). From x + y = a and y + z = a we can extract that y = |b - (x + y) - (y + z)| = |b - a - a| = |b - 2a| (b being the whole bottom side). Now since we've got y, we can easily calculate the other things too: x = (x + y) - y = a - y z = (z + y) - y = a - y

The answer:

x = a - y; y = |b - 2a|; z = a - y;