Summer Problem Solving Marathon Solution #2

The perpendicular from A to BC creates two 30-60-90 right triangles:



Because the two triangles ABD and ACD are congruent, BD and CD must both be 6:



In a 30-60-90 right triangle, the sides are in length ratios of 1 - √3 - 2. In the 30-60-90 right triangle ABD, the three sides in this ratio are BD, AD, and AB. Since BD is of length 6, AD must be of length 6√3 and AB of length 6*2 = 12 (which it is):



We now add point E at the midpoint of AD:



Since AD is of length 6√3 and E is its midpoint, ED is of length 3√3.

Now we add the line BE, which we want to find the length of:



This creates the right triangle BDE. We know that BD is of length 6, and DE of length 3√3. Using the Pythagorean Theorem, we then have:

BE =
√ 6^2 + (3√3)^2 

=
√ 36+27 

=
√ 63 

=3√7