What is the length of the diagonal of the square shown below?

1. Identify the shape and given information: The image shows a square with side lengths labeled as 5. A diagonal is drawn, dividing the square into two right-angled triangles. The angles within one of these triangles are shown as 45°, 45°, and 90°.

1. Recognize the properties of a square: A square has four equal sides and four right angles (90°). The diagonal of a square divides it into two congruent isosceles right-angled triangles.

3. Apply the Pythagorean theorem: The diagonal of the square is the hypotenuse of the right-angled triangle formed by two sides of the square. Let 'd' be the length of the diagonal and 's' be the length of a side of the square. According to the Pythagorean theorem, $a^2 + b^2 = c^2$, where 'a' and 'b' are the lengths of the legs and 'c' is the length of the hypotenuse.
In this case, the legs are the sides of the square, so $a = s$ and $b = s$. The hypotenuse is the diagonal, so $c = d$.
Therefore, the equation becomes $s^2 + s^2 = d^2$.

4. Substitute the given side length: The side length of the square is given as 5. Substitute $s = 5$ into the equation:
$5^2 + 5^2 = d^2$

5. Calculate the square of the side length:
$25 + 25 = d^2$

6. Sum the squared side lengths:
$50 = d^2$

7. Solve for the diagonal (d): To find the length of the diagonal, take the square root of both sides of the equation:
$d = \sqrt{50}$

8. Simplify the square root: The square root of 50 can be simplified by finding the largest perfect square factor of 50. The largest perfect square factor of 50 is 25.
$d = \sqrt{25 \times 2}$
$d = \sqrt{25} \times \sqrt{2}$
$d = 5\sqrt{2}$

Alternatively, using the property of a 45-45-90 triangle, the hypotenuse is $\sqrt{2}$ times the length of a leg. Since the legs are 5, the hypotenuse (diagonal) is $5\sqrt{2}$.