To solve for the side length of the base of the pyramid of Menkaure given that the diagonal of the base is 154 meters, let's utilize the properties of a square and the Pythagorean theorem. Here is a step-by-step method:
1. Understanding the relationship in a square:
- A square has four sides of equal length.
- The diagonal of a square splits it into two right-angled triangles.
2. Apply the Pythagorean Theorem:
- In each right-angled triangle formed by the diagonal, the two sides are equal (both are the side length of the square, let's call it [tex]\(a\)[/tex]), and the diagonal is the hypotenuse.
- By the Pythagorean theorem: [tex]\( \text{diagonal}^2 = \text{side}^2 + \text{side}^2 \)[/tex]
3. Set up the equation:
[tex]\[
d^2 = a^2 + a^2
\][/tex]
[tex]\[
d^2 = 2a^2
\][/tex]
4. Solve for the side length [tex]\(a\)[/tex]:
- Given that the diagonal [tex]\(d\)[/tex] is 154 m:
[tex]\[
154^2 = 2a^2
\][/tex]
- Calculate [tex]\(154^2\)[/tex] to get the value:
[tex]\[
154^2 = 23716
\][/tex]
[tex]\[
23716 = 2a^2
\][/tex]
- Divide both sides by 2:
[tex]\[
a^2 = \frac{23716}{2}
\][/tex]
[tex]\[
a^2 = 11858
\][/tex]
- Now, take the square root of both sides:
[tex]\[
a = \sqrt{11858}
\][/tex]
5. Calculate the value:
[tex]\[
a = \sqrt{11858} \approx 108.89444430272832
\][/tex]
6. Round to the nearest meter:
[tex]\[
a \approx 109 \text{ meters}
\][/tex]
Therefore, the side length of the base of the pyramid of Menkaure, rounded to the nearest meter, is 109 meters.
