Visualizing Cubes 1

The cube shown above has edges of length 2, and A and B are midpoints of the two edges. What is the length of AB (not shown)?

A. square root of 2

B. square root of 3

C. square root of 5

D. square root of 6

E. square root of 10

This question is intimidating, but with intimidating questions you must first gather what information is obvious and then proceed from what you have. All that you may notice now is that if A and B are the midpoints of the edges with lengths of 2, than the lengths from the closest corners to A and B must both have lengths of 1.

Now the question mentions that AB is not shown. All that must be done to show AB is draw a simple line with your pencil:

Wait, couldn't you makes triangle out of that? And the lengths of triangles are usually simples to find. In fact, you already know that the length from the corner to B=1.

Finding the length of A to that very same corner is a bit more complicated. You're going to need to use even more triangles. Notice how the floor of the cube is a square. You've drawn a line on that square from A to the upper corner. That square looks like this:

You know the length of A to the closer corner=1 because A is the midpoint of the side which equals 2, and you know that the length of the side of the square=2 because you were given that information.

Now it's time to use the Pythagorean theorem (a squared+b squared=c squared), which yields to us that the line from A to the corner= the square root of 5.

Now we refer back to the entire cube:

You now know the lengths of two sides of the right triangle (you know that is right because all the angles in a cube or square are right).

Now you must employ the Pythagorean theorem once more. The square root of 5 squared again is simply 5. 1 squared by itself is simply 1. The square root of the sum of these, 6, is the answer.

D's the correct answer.