06/17/2020 (2011: National Sprint Round, Problem 29)

Q: In pentagon ABCDE, BC = CD = DE = 2 units, ∠E is a right angle and m∠B = m∠C = m∠D = 135 degrees. The length of segment AE can be expressed in simplest radical form as a + 2√b units. What is the value of a + b?

A: 6

So we first draw a diagram and we get:

Then we try to divide this pentagon into triangles and/or quadrilaterals:

We use our knowledge to find the measure of the other angles:

We label everything that we can with the information given above and we get:

We continue labeling and get:

Which means that:

So AE = 2 + √2 + 2 + √2 = 4 + 2√2, which makes a = 4 and b = 2, so a + b = 4 + 2 = 6.