To find the area of a square when the length of its diagonal is given, you can follow these steps:
1. Identify the relationship between the diagonal and the side length of the square:
The diagonal of a square splits it into two right-angled triangles. Each of these triangles has sides of length equal to the side of the square and the hypotenuse equal to the diagonal. Let’s denote the side length of the square as [tex]\( s \)[/tex]. The mathematical relationship between the side length and the diagonal, [tex]\( d \)[/tex], in a square is given by the Pythagorean theorem:
[tex]\[
d = s \sqrt{2}
\][/tex]
2. Solve for the side length [tex]\( s \)[/tex]:
Given the diagonal length [tex]\( d \)[/tex] is 24 meters, we can isolate [tex]\( s \)[/tex] on one side of the equation:
[tex]\[
s = \frac{d}{\sqrt{2}}
\][/tex]
Substituting [tex]\( d = 24 \)[/tex]:
[tex]\[
s = \frac{24}{\sqrt{2}} \approx 16.97056274847714 \, \text{m}
\][/tex]
3. Calculate the area of the square:
The area [tex]\( A \)[/tex] of a square is given by the square of the side length:
[tex]\[
A = s^2
\][/tex]
Substituting [tex]\( s \approx 16.97056274847714 \)[/tex]:
[tex]\[
A = (16.97056274847714)^2 \approx 288 \, \text{m}^2
\][/tex]
So, the area of the square with a diagonal of 24 meters is approximately [tex]\( 288 \)[/tex] square meters.
None of the provided options (a. 1.88 m², b. 2 m, c. 2.88 m², d. 3 m) correctly match this solution. Hence, the correct area as justified by our calculations is approximately 288 m².
