To determine the length of the square’s diagonal given that the perimeter of the square is [tex]\(20 \sqrt{6}\)[/tex] cm, follow these steps:
1. Understand the relationship between the perimeter and the side length:
The perimeter [tex]\(P\)[/tex] of a square is given by [tex]\(4 \times \text{side length}\)[/tex]. Here, the perimeter is [tex]\(20 \sqrt{6}\)[/tex] cm.
2. Find the side length of the square:
We start with the equation for the perimeter:
[tex]\[
4 \times \text{side length} = 20 \sqrt{6}
\][/tex]
Solving for the side length, we divide both sides of the equation by 4:
[tex]\[
\text{side length} = \frac{20 \sqrt{6}}{4} = 5 \sqrt{6} \text{ cm}
\][/tex]
3. Determine the diagonal of the square:
The diagonal [tex]\(d\)[/tex] of a square can be found using the Pythagorean theorem in the context of the square's properties. Specifically, the diagonal forms the hypotenuse of a right triangle with the side lengths of the square. The relationship is given by:
[tex]\[
d = \text{side length} \times \sqrt{2}
\][/tex]
Substituting the side length we found:
[tex]\[
d = 5 \sqrt{6} \times \sqrt{2}
\][/tex]
Simplifying the expression under the square root:
[tex]\[
d = 5 \sqrt{6 \times 2} = 5 \sqrt{12} = 5 \times 2 \sqrt{3} = 10 \sqrt{3} \text{ cm}
\][/tex]
4. Numerical value of the diagonal:
For completeness, you can also compute the numerical value of [tex]\(10 \sqrt{3}\)[/tex]. Knowing [tex]\(\sqrt{3} \approx 1.732\)[/tex]:
[tex]\[
d \approx 10 \times 1.732 \approx 17.32 \text{ cm}
\][/tex]
Thus, the length of the square's diagonal is approximately [tex]\(17.32\)[/tex] cm.
