Apply the Pythagorean Theorem to determine the correct formula for finding the length of the diagonal of the cube.

Question

(1 point)
Responses

d2−x2−x2−−−−−−−−−−√
𝑑
2
−
𝑥
2
−
𝑥
2
start root d squared minus x squared minus x squared end root

a2−x2−x2−−−−−−−−−−√
𝑎
2
−
𝑥
2
−
𝑥
2
start root a squared minus x squared minus x squared end root

a2+x2+x2−−−−−−−−−−√
𝑎
2
+
𝑥
2
+
𝑥
2
start root a squared plus x squared plus x squared end root

x2+x2+x2−−−−−−−−−−√

Answer 1

To find the length of the diagonal of a cube using the Pythagorean Theorem, we can consider a cube with side length $x$.

1. First, find the diagonal of one face of the cube: For one face (which is a square), the diagonal $d_1$ can be found using the Pythagorean Theorem: $$d_1=\sqrt{x^2+x^2}=\sqrt{2x^2}=x\sqrt{2}$$

2. Next, consider the diagonal that goes through the cube: Now, to find the diagonal $d$ of the cube itself, we treat it as a right triangle formed by the diagonal of the face $d_1$ and the third dimension of the cube: $$d=\sqrt{(d_1{)}^2+x^2}=\sqrt{(x\sqrt{2}{)}^2+x^2}$$ $$d=\sqrt{2x^2+x^2}=\sqrt{3x^2}=x\sqrt{3}$$

Therefore, the formula for the length of the diagonal of a cube with side length $x$ is: $$d=x\sqrt{3}$$

Now looking at the provided responses, the correct formulation for finding the length of the diagonal $d$ of the cube in the form of the Pythagorean theorem is: $$\sqrt{x^2+x^2+x^2}\text{which is simply }\sqrt{3x^2}\text{ or the response }{x}^2+x^2+x^2\sqrt{}$$

So the best response aligning closely with this derived formula is: $\sqrt{x^2+x^2+x^2}$.