Try this free sample question!

In the cube above, the length of each edge is 3.
What is the straight-line distance from vertex A to vertex B?

(A) 3
(B) 2 √3
(C) 3 √2
(D) 3 √3
(E) 9

Answer & Explanation

A good first step is to simply draw the line that shows the distance from point A to point B. Thinking in three dimensions will not only be important in answering this question, but will also be challenging. Nearly every time you need to find the distance between two points in a figure you should try to make a right triangle that has that distance as the hypotenuse.

Next, try to create a right triangle that has the distance of AB as the hypotenuse. You can do this by drawing a line from A directly down to the vertex below, then diagonally across the base of the cube to meet B. Though perspective makes it tricky, this triangle is the one you need. Use the Pythagorean Theorem to determine the length of AB. Each edge of the cube equals 3, so add this information to your figure. This gives you one side of this triangle.

You now need to find the distance across the base of the cube. If you look carefully, you'll notice that you have another right triangle on the base of the cube with lengths of 3.

Given that the diagonal of this square gives us a 45-45-90 triangle, you can use X : X : X√2 to determine the length of this diagonal, which is also the hypotenuse of the triangle on the base of the cube. Since each of the lengths of this triangle are 3, the hypotenuse must be 3√2, which means the base length of our original triangle is also 3√2.

Now that you know the two shorter lengths of the original triangle with hypotenuse AB, you can determine the length of AB by using the Pythagorean Theorem.

Simplifying the formula gives you: 9 + 18 = (AB)². You can now square each side to find that AB = √27, which can then be reduced to 3√3. Based on this, the correct answer is D.

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