From the course: Preparing for the GMAT

Special right triangles

From the course: Preparing for the GMAT

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Special right triangles

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- [Narrator] There are a couple of right triangles that we call Special Right Triangles because their sidelines always have the same ratio. These are the 45-45-90 right triangle and the 30-60-90 right triangle. So let's take a look at a couple of examples of these triangles in action. Let's start with the 45-45-90 right triangle. So we call it 45-45-90 because that angle is 45, so is that, and then of course the right angle there is 90 degrees. What do we know about this triangle right off the bat? Well, we know that these sides are equal, right, because it's an isosceles triangle. It's got two equal angles, therefore the sides opposite those equal angles are also equal to each other. So there's a formula here to know about the 45-45-90 right triangle. If we call the legs of that triangle "X", and we can because they're equal, the hypotenuse would be X times root two. So let's see, what if we had a problem and we said to ourselves, well here's a 45-degree angle and here's a 45-degree angle. The nice thing about these Special Right Triangles, you only need to know one side to solve for the other two. So what if we somehow knew that the hypotenuse was three times root two? We should be able to immediately fill in that each leg is three, because of this formula over here. We know that the hypotenuse is just the leg times root two. OK, so this is our 45-45-90 right triangle. Now let's talk about a 30-60-90 right triangle. So, as you can probably tell, this is just a right triangle with 30-60-90 as its angles. So if that's our right angle and that's a 30-degree angle, therefore that must be a 60-degree angle, because of course in the triangle, all the angles add up to 180. There's a formula here to learn as well. The short is side X, or the short leg is X; the hypotenuse is always 2X, if those are the angle measures, and then the side opposite the 60-degree angle is going to be X times the square root of three. So this might come into play, let's see, if we have a triangle and let's say we know that this is 60 and this is 30, that's a right angle, maybe somebody tells us that this is eight. We should be immediately able to fill in that that's four, and that's four root three. Why? Because of this formula. Special right triangles behave the same way every time. So, then if know that this is a 30-60-90 right triangle, the short side opposite the 30-degree angle is always going to be half of the hypotenuse and then the middle side is going to be the short side times root three. So make sure you know the formulas for these triangles. They are super useful and they tend to pop up pretty consistently, and the nice thing is you only need one side to solve for the other two sides.

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