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Square \(ABCD\) has side length 1 unit. Points E and F are on sides AB and CB, respectively, with AE = CF. When the square is folded along the lines DE and DF, sides AD and CD coincide and lie on diagonal BD. The length of segment AE can be expressed in the form \(\sqrt{k}-m\) units. What is the integer value of K + m?

 Mar 15, 2019
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Ok, got it. The diagonal of the square is an angle bisector, which means that ADB = 45 degrees. Using a little common sense, ADE is an angle bisector as well. How I figured this out was if the angle wasn't equal on both sides, then the square sides would overlap. So ADE = 22.5 degrees. I used trig on this part, but Tan 22.5 = 0.41421356237 about. So then I got a suspicion, so I plugged in \(\sqrt 2 -1\) into the calculator, and I got the same answer. So the answer is \(\sqrt 2 -1\) , which you need to find the sum, so 2 + 1 = 3. Sorry my method is a little unorthodox, but it's the best I could come up with. I tried doing something with the angle bisector theorem, and I got \(\frac {1}{\sqrt2} = \frac{AE}{EB}\), but I couldn't get any further than that.

 

Hope this helps!

 Mar 15, 2019
edited by LagTho  Mar 15, 2019
edited by LagTho  Mar 15, 2019

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