A critical aspect of GMAT Test Prep is the study of Geometry – especially of areas where you might be rusty or very unfamiliar. For me when I was studying for the GMAT, one area where I was very rusty was the study of diagonals. Watch the video below to learn more; we also include the transcript for your convenience.
GMAT Test Prep: 3D Diagonal Review [Transcript]
Please pause this video and answer the question before continuing with GMAT Prep Geometry. Now in this question, we want to find the distance between points A and B. We will solve this question using two different approaches. For the first solution, let’s begin by looking inside the box to get a better idea of what the question is asking. So we want to find the length of AB. We’re going to find this length in two steps.
First, we will find the length of this diagonal along the base of the box, and then we will combine this line segment with the height of the box and the line segment AB to create a right triangle. At this point, we will be able to find the length of AB. So, let’s begin with this diagonal along the base of the box. Notice that we have a right triangle here. So if we take a closer look at this triangle, we can see that we have the lengths of two sides and we want to find the length of the hypotenuse.
To find this length, we could apply the Pythagorean theorem, or we could recognize that this triangle is a multiple of the three, four, five triangle we examined in an earlier lesson. Since the purple triangle is twice as large as the three, four, five triangle, the hypotenuse here must be 10. Now when we add this information to our diagram, we can see that we have completed the first step in this solution. At this point, we will highlight the height of the rectangle and recognize that these two sides meet at a 90 degree angle. So when we add the line segment AB, we can see that we have a right triangle and side AB is the hypotenuse of that triangle. Let’s call side AB, X and then take a closer look at this triangle over here.
Now, before we apply the Pythagorean theorem to find the value of X, we should take a moment to determine whether this triangle is a multiple of one of our Pythagorean triples in order to save some time. Now since this is not a multiple, we will apply the Pythagorean theorem. So we will first plug in the lengths of the two legs and the hypotenuse. Then we will evaluate 10 squared and 7 squared and add these two values. If X squared is equal to 149, then X must equal the square root of 149, which means the answer here is D.
Okay, now let’s look at a nice formula that handles these types of three dimensional diagonals. It goes like this. If we have a rectangular box with dimensions W, X and Y, then the distance from point A to point B is equal to the square root of the sum of the squares of the dimensions. So in this particular example, the length of AB is equal to the square root of eight squared, plus six squared, plus seven squared. When we evaluate and simplify this, we get the square root of 149. So the answer is still D.