Primary school competition problem find the area of a square

Someone posted online a primary school competition problem. I solved it using the Cayley-Menger determinant (finding the answer is 169/2 after getting the length of AD being 13) but it probably isn't the intended solution. I cannot find a "primary school" way to solve it. The problem is to find the area of the square given the following:

The angles $$\angle ABC=90^o,\angle BCD=90^o$$.

By the Pythagorean theorem (the diagram can be rotated such that the interior segments are all horizontal/vertical), the diagonal $$DA$$ has length $$\sqrt{12^2+5^2}=13$$, so the square side length is $$\frac{13}{\sqrt2}$$ and the area $$\frac{169}2$$.