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).
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$\begingroup$ The parallelogram isn't uniquely defined if you only know $a$ and $b$. $\endgroup$– Tc14Commented Apr 29, 2020 at 19:10
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$\begingroup$ @Micah OP wants to find $x$, $y$, and $z$ as labeled in the image. $\endgroup$– Tc14Commented Apr 29, 2020 at 19:12
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$\begingroup$ Yes. If You need, You can substitute a = 3 and b = 8. Those are the things that are given $\endgroup$– Samuel SmithCommented Apr 29, 2020 at 19:13
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$\begingroup$ @Tc14Hd My bad, I was trying to figure it out from "calculate the opposite side of the parallelogram..." $\endgroup$– Micah WindsorCommented Apr 29, 2020 at 19:14
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1$\begingroup$ @AlejandroBergasaAlonso Yes, thank You! I hadn't noticed that. I can probably solve it now. It's such an obvious thing to miss that it's quite embarrassing... :/ $\endgroup$– Samuel SmithCommented Apr 29, 2020 at 19:18
3 Answers
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, $x=b-a$.
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 $y = b - 2a$.
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;