The shortest distance from one vertex of a cube to the opposite vertex (also known as the space diagonal) can be found using the Pythagorean theorem in three dimensions.

Here are the steps:

1. Let's assume the side length of the cube is 'a'.

2. The diagonal of one face of the cube can be found using the Pythagorean theorem in two dimensions: a^2 + a^2 = d^2, where d is the diagonal of one face. This simplifies to 2a^2 = d^2, so d = sqrt(2a^2) = a*sqrt(2).

3. Now, we can find the space diagonal (the shortest distance from one vertex to the opposite vertex) by applying the Pythagorean theorem again, this time in three dimensions: a^2 + d^2 = s^2, where s is the space diagonal. Substituting the value of d from step 2, we get a^2 + (a*sqrt(2))^2 = s^2. This simplifies to a^2 + 2a^2 = s^2, so 3a^2 = s^2.

4. Therefore, the shortest distance the ant can take to reach the opposite vertex of the cube is s = sqrt(3a^2) = a*sqrt(3).

Question

The shortest distance from one vertex of a cube to the opposite vertex (also known as the space diagonal) can be found using the Pythagorean theorem in three dimensions.

Here are the steps:

1. Let's assume the side length of the cube is 'a'.

2. The diagonal of one face of the cube can be found using the Pythagorean theorem in two dimensions: a^2 + a^2 = d^2, where d is the diagonal of one face. This simplifies to 2a^2 = d^2, so d = sqrt(2a^2) = a*sqrt(2).

3. Now, we can find the space diagonal (the shortest distance from one vertex to the opposite vertex) by applying the Pythagorean theorem again, this time in three dimensions: a^2 + d^2 = s^2, where s is the space diagonal. Substituting the value of d from step 2, we get a^2 + (a*sqrt(2))^2 = s^2. This simplifies to a^2 + 2a^2 = s^2, so 3a^2 = s^2.

4. Therefore, the shortest distance the ant can take to reach the opposite vertex of the cube is s = sqrt(3a^2) = a*sqrt(3).

Knowee AI · Accepted Answer

The shortest distance from one vertex of a cube to the opposite vertex (also known as the space diagonal) can be found using the Pythagorean theorem in three dimensions.

Here are the steps:

1. Let's assume the side length of the cube is 'a'.

2. The diagonal of one face of the cube can be found using the Pythagorean theorem in two dimensions: a^2 + a^2 = d^2, where d is the diagonal of one face. This simplifies to 2a^2 = d^2, so d = sqrt(2a^2) = a*sqrt(2).

3. Now, we can find the space diagonal (the shortest distance from one vertex to the opposite vertex) by applying the Pythagorean theorem again, this time in three dimensions: a^2 + d^2 = s^2, where s is the space diagonal. Substituting the value of d from step 2, we get a^2 + (a*sqrt(2))^2 = s^2. This simplifies to a^2 + 2a^2 = s^2, so 3a^2 = s^2.

4. Therefore, the shortest distance the ant can take to reach the opposite vertex of the cube is s = sqrt(3a^2) = a*sqrt(3).

An ant is at the center of a cube. What is the shortest distance it can take to reach the opposite vertex of the cube?

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