Sure, let's solve the problem step-by-step to find the diagonal of a square whose side length is given as [tex]\(7 \sqrt{3}\)[/tex].
1. Express the given side length:
The side length of the square is [tex]\(7 \sqrt{3}\)[/tex].
2. Formula for the diagonal of a square:
The formula to calculate the diagonal [tex]\(d\)[/tex] of a square with side length [tex]\(a\)[/tex] is given by:
[tex]\[
d = a \cdot \sqrt{2}
\][/tex]
where [tex]\(a\)[/tex] is the side length of the square.
3. Substitute the side length into the formula:
In this case, our side length [tex]\(a\)[/tex] is [tex]\(7 \sqrt{3}\)[/tex]. So,
[tex]\[
d = 7 \sqrt{3} \cdot \sqrt{2}
\][/tex]
4. Simplify the expression:
Combine the square roots:
[tex]\[
\sqrt{3} \cdot \sqrt{2} = \sqrt{6}
\][/tex]
Now multiply:
[tex]\[
d = 7 \cdot \sqrt{6}
\][/tex]
5. Calculate the numerical value:
Using the value [tex]\( \sqrt{6} \approx 2.449 \)[/tex]:
[tex]\[
d \approx 7 \cdot 2.449 = 17.146
\][/tex]
So the diagonal of the square whose sides are [tex]\(7 \sqrt{3}\)[/tex] is approximately [tex]\(17.146\)[/tex].
The numerical results for the side length and the diagonal are:
- Side Length: [tex]\(7 \sqrt{3} \approx 12.124\)[/tex]
- Diagonal: [tex]\(17.146\)[/tex]
These values confirm that the calculations are consistent with the correct step-by-step solution, ensuring our understanding of the problem and the mathematical process.
