This video will be a review of right triangles and show you how they’re used on the GMAT Exam.
GMAT Test Review on Right Triangles [Transcript]
Instructor: In this lesson we will examine some properties of right triangles. To begin, a right triangle is a triangle that has a 90° angle as one of its angles. The two sides that intersect to create the right are called legs and the remaining side is called the hypotenuse. The hypotenuse is always the longest side of a right triangle. This should make sense to us because it is the side that is opposite the 90° angle. Since the 90° angle will always be the largest angle in a right triangle and since we have a rule that says the size of a triangle corresponds to their opposite angles, we can conclude that the hypotenuse must be the longest side. One of the interesting properties of right triangles is that for any right triangle if we square the length of one leg and add it to the square of the other leg, the sum will always equal the square of the hypotenuse. This property is known as the Pythagorean theorem and it is typically expressed as follows: if we have a right triangle with sides a, b, and c where c is the hypotenuse, then a squared plus b squared equals c squared.
Keep in mind that this theorem works both ways. That is if we have a right triangle then it must be true that the lengths are such that a squared plus b squared equals c squared. And, if we have a triangle with lengths such that a squared plus b squared equals c squared then that triangle must be a right triangle. Okay, now let’s practice applying the Pythagorean theorem. In this question, we want to find the missing length x. Since we have a right triangle here we can apply the Pythagorean theorem. So if we square the length of both legs and add them together this will equal the hypotenuse x squared. To solve for x, we will evaluate eight squared and six squared and add them together. If x squared is equal to 100 then x must equal the square root of 100 which is 10. So the length of the missing side here is 10.
Let’s try another example. Once again, we have a right triangle. So we will call up the Pythagorean theorem. This time, the missing length is one of the legs. In the first example, the missing length was the hypotenuse. So to apply the theorem, we will first square both legs, x and four, and then add them together. This will equal the hypotenuse six squared. To solve for x, we will evaluate four squared and six squared and then subtract 16 from both sides. If x squared is equal to 20 then x must equal the square root of 20. At this point, we can simplify the square root of 20 by first rewriting it as root four times root five. And since the square root of four is two, we can see that x is equal to two times root five.
Okay, now let’s take a closer look at these two right triangles. Notice that the lengths of the first triangle are all integers. But in the second triangle, one of the lengths is two root five, which is not an integer. We are particularly interested in right triangle such as this where the lengths of all three sides are integers. Here the lengths are 6, 8, and 10 make up something called a Pythagorean triple. A Pythagorean triple is a set of three integers, or whole numbers, that can be the size of a right triangle. These triples are important because they show up frequently on tests. The most common Pythagorean triple is 3-4-5 which represents a right triangle with lengths 3, 4, and 5. Notice that these three integers nicely satisfy the Pythagorean theorem. Another popular Pythagorean triple is the 5-12-13 triangle. And once again, notice that the three numbers here satisfy the Pythagorean theorem. Two other triples to keep in mind are 8-15-17 and 7-24-25. But, these triples do not show up often on tests.
If you can memorize these triples you can save yourself considerable time on test day. For example, it would typically take a lot of time to find the length of the missing side here, if we are forced to apply the Pythagorean theorem. We would have to solve the subsequent equation for x which could take a while. However, if we recognize that we’ve been given two sides of an 8-15-17 right triangle then the last remaining side must have length eight. So be sure to memorize these Pythagorean triples before test day. You should also watch out for multiples of these triples. For example, if we take the size of a 3-4-5 right triangle and double them we get a 6-8-10 right triangle, which is another Pythagorean triple. Similarly, if we triple the lengths of a 3-4-5 triangle, we get another triple. In fact, the multiples of any of these Pythagorean triples will also be Pythagorean triples.
Okay, now let’s see how the multiples of Pythagorean triples can help us. Here, we want to find the length of the missing side. Now we could apply the Pythagorean theorem here but 28 and 35 are very large numbers to work with. Alternatively, we should be looking for multiples of Pythagorean triples. This triangle appears to be a multiple of a 3-4-5 right triangle. Notice that this side is seven times as long as the side with length four and this side is seven times as long as the side was length five. We can see that this big triangle here is seven times as large as the 3-4-5 triangle. As such, the last side will be seven times as long as the side with length three. So the missing side has length 21.
Now an important point to mention here is that if you want to use Pythagorean triples you must have two corresponding sides. Here’s what I mean. Let’s say that we have a triangle with the hypotenuse with length 50 and we want to find the lengths of the other two sides. Well, we might conclude that the large triangle is a multiple of a 3-4-5 triangle. And since the large triangle has hypotenuse 50 and the small triangle has hypotenuse five, we might further conclude that the large triangle is 10 times as large as the small triangle. If we make this conclusion, then we will also conclude that the two missing sides are 10 times as long as their corresponding sides. Now all of these conclusions are misguided. Notice that we could’ve also concluded that the big triangle is a multiple of the 7-24-25 triangle. And since 50 is two times as large as 25, we could have concluded that the large triangle is two times as large as the small triangle, in which case the two missing sides would have lengths 14 and 48. So as you can see, having just one length in a right triangle is not enough information to begin looking for Pythagorean triples. We need two corresponding sides in order to use these triples.
Now before we examine one more example, please notice that the word corresponding here is underlined. The reason for this is that it is not enough to simply have two sides before reaping the benefits of Pythagorean triples. Those two sides must be corresponding sides. Here’s what I mean. Here we have a right triangle and the two given sides have length three and four. At this point, we might assume that we have a 3-4-5 right triangle here, in which case, the length of the missing side must be five. However, there is a problem here. The problem is that in the original Pythagorean triple, the hypotenuse has length five. And in our example triangle, the hypotenuse has length four. Also, notice that the hypotenuse in the example triangle is not the longest side, even though the hypotenuse must always be the longest side in a right triangle. So for these reasons, we can see that the missing side here does not have length five. So to use the Pythagorean triple 3-4-5 in this example, the corresponding sides must match up. Since the lengths of the hypotenuses do not match up we cannot use this triple.
So how do we find the missing side then? Well, we still have a right triangle so we can apply the Pythagorean theorem. When we plug in the values, we get an equation that we can solve for x. First, we will evaluate three squared and four squared, then we will subtract nine from both sides to see that x squared equals seven. If x squared equals seven, then x must equal the square root of seven. So the missing side has length root 7. Okay, to summarize: in this lesson we learn how to apply the Pythagorean theorem and we learned how Pythagorean triples can save us a lot of time when used correctly.