Finding the greatest possible dimension of a rectangle

Question

Here's the question:

If the greatest possible distance between two points within a certain rectangular solid is 12, then which of the following could be the dimension of the solid?

A) 3x3x9 B) 3x6x7 C) 3x8x12 D) 4x7x9 E) 4x8x8

I set up the problem by attaching a point from one vertex of the prism (on the most top-left ) to the bottom vertex of the prism (on the most bottom-right) and made it equal to 12. However, I get very confused with the different Pythagorean equations I have to go through. Help?

Answer 1

You want $12 = \sqrt{x^2+y^2 + z^2}$, where $x,y,z$ are the three side lengths. Alternatively, you want $x^2+y^2+z^2 = 144$

As an example, note that $3^2+3^2+9^2 = 99$, so A) is not correct.

Answer 2

We are used to 2-dimensionl Pythagoras: $d=\sqrt{x^2+y^2}$

For this problem you need to use 3-dimensionl Pythagoras: $d=\sqrt{x^2+y^2+z^2}$