Question

1. find the perimeter of a square with a diagonal of 12 centimeters.

Explanation:

Step1: Use Pythagorean theorem for square.

Let the side - length of the square be $a$. In a square, if the diagonal is $d$, by the Pythagorean theorem $d^{2}=a^{2}+a^{2}=2a^{2}$. Given $d = 12$ cm, so $12^{2}=2a^{2}$, which is $144 = 2a^{2}$.

Step2: Solve for the side - length $a$.

Divide both sides of the equation $144 = 2a^{2}$ by 2: $\frac{144}{2}=a^{2}$, so $a^{2}=72$, and $a=\sqrt{72}=6\sqrt{2}$ cm.

Step3: Calculate the perimeter $P$ of the square.

The perimeter of a square is $P = 4a$. Substitute $a = 6\sqrt{2}$ into the formula, we get $P=4\times6\sqrt{2}=24\sqrt{2}$ cm.

Answer:

$24\sqrt{2}$ cm