SOLUTION: A point M is chosen inside the square ABCD in such a way that angleMAC=angleMCD=x. Determine angleABM in terms of x.

Question 1120937: A point M is chosen inside the square ABCD in such a way that angleMAC=angleMCD=x. Determine angleABM in terms of x.

Answer by greenestamps(13206) (Show Source): You can put this solution on YOUR website!

The figure looks something like this (not to scale)....

The given condition makes angle MCA (45-x) degrees; with angle MAC x degrees, we know that angle AMC is 135 degrees.

Knowing that, it seems there should be a clever way to draw some auxiliary lines in the figure to come up with a relatively simple solution using only geometry.

But I have not been able to find that path.

Perhaps another tutor will see a nice way to derive such a simple solution....

Empirically, the answer is that angle ABM (y in my figure) is (90-2x) degrees -- i.e., angle CBM is 2x degrees.

I come to that conclusion by looking at three special cases:

(1) In the limiting case where x is 0 degrees, point M is coincident with point C, and angle y is 90 degrees.
(2) In the limiting case where x is 45 degrees, point M is coincident with point A, and angle y is 0 degrees.
(3) And in the special case where x is 22.5 degrees, point M will be on diagonal BD, and angle y is 45 degrees.

So we have three (x,y) data points: (0,90), (22.5,45), and (45,0).

A simple relationship between those three data points is y=90-2x.
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