figure it out

My answer is between the brackets below. Highlight to see.

[We're given three squares of different sizes that are stacked in a certain way. The two smaller squares have areas of 4 and 36 units respectively, and they're stacked on each other with the smaller square on top and the two squares' left sides collinear. The third and largest square is stacked such that its "western" corner is the same point as the smaller square's "northeastern" corner, and the large square's side is also collinear with Area = 36 square's northeastern corner. The largest square's "southern" corner is collinear with the 36 square's south-facing side. Our job is to determine the length of the diagonal AB of the largest square. 

Ready, Poison Girls?

The first and most obvious step is to note the side lengths of the two smaller squares. For the Area = 4 square, the side length is 2. For the Area = 36 square, the side length is 6. The length of the "north" side of the Area = 36 square is 6, so for the line segment going from the smallest square's "southeastern" corner to the mid-sized square's "northeastern" corner, the segment's length is 4 because 6 - 2 = 4.

Note that we can see several right triangles in the picture. Let's forget about the two big, obvious right triangles inside the largest square and concentrate on the two smaller right triangles formed by the "negative" space defined by the largest square's "southwestern" side and the two smaller squares' "eastern" sides plus the horizontal segment with a length of 4 and that other horizontal segment defined by the mid-sized square's "southeastern" corner and Point A, a segment of undetermined length. 

Note that these two right triangles are similar (angle-angle-angle similarity, as deduced from parallel line segments). So, to find the length of the "undetermined" horizontal segment, we set up a proportion based on triangle similarity:

2/4 = 6/x

We cross-multiply, solve for x, and realize x = 12. 

We're close to figuring out the side length of the largest square, which is the sum of the hypotenuse lengths of the two right triangles in question.

The Pythagorean Theorem must now be broken out to solve for these two hypotenuses. The basic formula is a² + b² = c².

First triangle: 2² + 4² = c², so c = 2√5

Second triangle: 6² + 12² = c², so c = 6√5

Add these together, and the side length of the biggest square is 8√5.

We recall that the length of a square's diagonal equals √2 times the length of the side, so the diagonal's length is 8√10, or in approximate decimals, 25.298.

QED.]