Question

1. the area of a square is 81 cm². what is the length of the diagonal of the square? express your answer in simplest radical form.

Explanation:

Step1: Find side - length of square

Let the side - length of the square be $s$. The area formula of a square is $A = s^{2}$. Given $A = 81\ cm^{2}$, then $s^{2}=81$, so $s=\sqrt{81}=9\ cm$.

Step2: Use Pythagorean theorem for diagonal

In a square, if the side - length is $s$ and the diagonal is $d$, by the Pythagorean theorem $d^{2}=s^{2}+s^{2}$. Since $s = 9\ cm$, then $d^{2}=9^{2}+9^{2}=81 + 81=162$.

Step3: Solve for diagonal

$d=\sqrt{162}=\sqrt{81\times2}=\sqrt{81}\times\sqrt{2}=9\sqrt{2}\ cm$.

Answer:

$9\sqrt{2}\ cm$