Square Diagonal Formula: Finding Length in Terms of Side 'a'

To find the length of the diagonal of a square with side length $a$, we will use the Pythagorean theorem.

A diagonal divides a square into two congruent right-angled triangles. The legs of these triangles are the sides of the square, both with length $a$. The diagonal then serves as the hypotenuse.

According to the Pythagorean theorem, the hypotenuse $c$ of a right triangle with legs $a$ is found via:

• $c = \sqrt{a^2 + a^2}$

Simplifying inside the square root gives us:

• $c = \sqrt{2a^2}$

We can further simplify this expression:

• $c = \sqrt{2} \cdot \sqrt{a^2} = \sqrt{2}a$

Thus, the length of the diagonal is expressed by the function $y = \sqrt{2}a$.

In this problem, this solution corresponds to choice 1, which is $y = \sqrt{2}a$.

The chosen answer is correct as verified by the application of the Pythagorean theorem to a square.

Therefore, the correct function for the diagonal is $y = \sqrt{2}a$.