Question

if the length of a diagonal in a square is a, what is the length of its side?

Explanation:

Step1: Recall Pythagorean theorem

Let the side - length of the square be $x$. In a square, using the Pythagorean theorem for the right - triangle formed by two adjacent sides and a diagonal. For a right - triangle with legs of length $x$ and hypotenuse of length $a$, we have $x^{2}+x^{2}=a^{2}$.

Step2: Simplify the equation

Combining like terms, $2x^{2}=a^{2}$. Then $x^{2}=\frac{a^{2}}{2}$.

Step3: Solve for $x$

Taking the square root of both sides, $x = \frac{a}{\sqrt{2}}=\frac{\sqrt{2}a}{2}$ (we take the positive value since length cannot be negative).

Answer:

$\frac{\sqrt{2}a}{2}$