Question

What is the longest line segment that can be drawn in a right rectangular prism that is 13 cm long, 8 cm wide and 7 cm tall?

Answer 1

Look at the attached picture. The longest segment we can draw is AG (or any of its equivalent: BH, CE, DF). Let's focus on AG for example.

We can think of AG as the hypotenuse of triangle ACG. So, we need AC first. AC is itself the hypotenuse of triangle ABC, so we have

$AC^2 = AB^2+BC^2=7^2+8^2=113$

Now we can resume where we stopped with triangle AGC, and we have

$AG=\sqrt{AC^2+CG^2}=\sqrt{113+169}=\sqrt{282}$