Question

write the area a of a square with a side of length s as a function of its diagonal, d. hint: write down the formulas involving the area, perimeter, and diagonal of a square. the formula for the area of the square as a function of its diagonal is a = (simplify your answer.)

Explanation:

Step1: Recall Pythagorean - theorem for square

In a square with side length $s$ and diagonal $d$, by the Pythagorean theorem, $s^{2}+s^{2}=d^{2}$.

Step2: Simplify the Pythagorean - relation

Combining like terms, $2s^{2}=d^{2}$, then $s^{2}=\frac{d^{2}}{2}$.

Step3: Recall the area formula of a square

The area of a square $A = s^{2}$.

Step4: Substitute $s^{2}$

Since $s^{2}=\frac{d^{2}}{2}$, then $A=\frac{d^{2}}{2}$.

Answer:

$\frac{d^{2}}{2}$