Elegant Solutions vs. Hacking

This is a study in contrasts around a fun problem by @eylem:

An elegant solution to this would be as follows:

• After angle chasing to find angle DEA is 45 degrees extending DE to make a full right isosceles triangle with side lengths of 15 sqrt(2)/2. 
• Use the Pythagorean Theorem to find the missing side length of AGE. This gives you the base of the triangle DE also,
• Then note AGE is congruent to the halves of CDEs so you also know the altitude.
• Apply the triangle area formula.

But there are other shadier options for solving the problem:

Instead, we can take advantage of patterns in the values of the lengths and the then use knowledge about 3-4-5 triangles (see: 12-triangles-and-their-link-to-pythagorean-triples) to crack the problem.

This actually exposes a bit of structure that was not seen in the first solution.  If you didn’t think to try the educated initial guess, you could more directly find it by setting up a simple quadratic equation based on the Pythagorean Theorem:

\[(5\sqrt{5})^2 + (15 - x)^2 = x^2\]