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In pentagon $ABCDE$, $BC=CD=DE=2$ units, $\angle E$ is a right angle and $m \angle B = m \angle C = m \angle D = 135^\circ$. The length of segment $AE$ can be expressed in simplest radical form as $a+2\sqrt{b}$ units. What is the value of $a+b$?

 Mar 3, 2021
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If we draw a line starting from B that is orthogonal to BC going down to AE at point Fand a line that starts at C that is orthogonal to BC as well but only goes to the same height as D at point G and connect the two points, we get two new 45-45-90 triangles, and AF+BC+GD=AE, and AF=BF, and BF=CG+DE=sqrt2+2, so sqrt2+2+2+sqrt2=2sqrt2+4=AE, so a=4, and b=2, so a+b=$\boxed{6}$

 Mar 3, 2021

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