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: \[ \sqrt{x^2 + x^2 + x^2} \quad \text{which is simply } \sqrt{3x^2} \text{ or the response } x^2 + x^2 + x^2\sqrt{} \]

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