To find the length of each side of a square given its diagonal, we need to use the Pythagorean theorem. The diagonal of a square divides it into two right-angled triangles, where the legs of the triangles are the sides of the square, and the hypotenuse is the diagonal.
Let's denote the side length of the square as [tex]\( s \)[/tex].
1. According to the Pythagorean theorem:
[tex]\[
s^2 + s^2 = (\text{diagonal})^2
\][/tex]
Simplifying, we get:
[tex]\[
2s^2 = (\text{diagonal})^2
\][/tex]
2. The given diagonal length is 12 inches. Plugging this into the equation:
[tex]\[
2s^2 = 12^2
\][/tex]
[tex]\[
2s^2 = 144
\][/tex]
3. Now, solve for [tex]\( s^2 \)[/tex]:
[tex]\[
s^2 = \frac{144}{2}
\][/tex]
[tex]\[
s^2 = 72
\][/tex]
4. Taking the square root of both sides to find [tex]\( s \)[/tex]:
[tex]\[
s = \sqrt{72}
\][/tex]
[tex]\[
s = \sqrt{36 \times 2}
\][/tex]
[tex]\[
s = 6\sqrt{2}
\][/tex]
Therefore, the length of each side of the square MNOP is [tex]\( 6 \sqrt{2} \)[/tex] inches. The correct answer is:
[tex]\[
6 \sqrt{2}
\][/tex]
