The diagonal of the square creates two congruent right triangles, which you could see if you drew a picture. The diagonal is the hypotenuse of the triangle, and the sides of the square are the legs of the triangle. Again, a diagram might help.
The pythagorean theorem is (a^2)+(b^2)=(c^2), where c is the hypotenuse and a and b are the legs.
We know that c is 5 square root of 2, so:
(a^2)+(b^2)=((5 square root of 2)^2),
Now, distribute the square (exponent of 2) to both the 5 and the square root of 2. Squaring and the square root cancel each other out, leaving us with 2. 5^2 is 25. Then, both of those are multiplied together, so:
(a^2)+(b^2)=50
Since we are dealing with a square, both side lengths are the same, so a and b are the same number. So, we have two of the same term being added to each other. To eliminate any confusion, let x stand for the length of the sides of the triangle. This is equivalent to:
2(x^2)=50.
Then, we just solve for x.
(x^2)=25
x=5
All sides of the triangle are 5. So, the area is 5*5, or 25 inches.