Special Right Triangles- Types and Properties
Not all right triangles are the same. In this lesson, we'll look at two special right triangles (30-60-90 and 45-45-90) that have unique properties to help you quickly and easily solve certain triangle problems.
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Special Right Triangles
Triangles are like people. Some are just special in certain ways. For example, maybe you’re a great tennis player. You might be the star of your high school’s varsity team. That would make you a special tennis player, certainly better than most people at the game. That’s like the difference between a right triangle and an oblique triangle. A right triangle probably has a wicked serve.
And then there are special right triangles. These aren’t your regional tennis champions. These are your Wimbledon champions. In this lesson, we’re going to focus on two right triangle equivalents of tennis royalty.
The first special right triangle is the 30-60-90 triangle . I know it sounds like a cell phone plan, but 30-60-90 refers to the triangle’s angles.
You know that the side opposite the right angle is the hypotenuse. We call the side opposite the 30-degree angle the short leg. The one opposite the 60-degree angle? That’s the long leg. Not too special, but pretty easy to remember.
So, what is special? The ratio of the side lengths. If the hypotenuse is 2, the short leg is 1, and the long leg is the square root of 3.
So, how do you remember which is which? Just think about the logical relative length of the sides. The short leg is the shortest side, so that’s your 1. And the long leg is opposite the 60-degree angle, which makes it the middle one, so that’s your square root of 3. Finally, the hypotenuse is the longest, and it’s opposite the largest angle, so that’s your 2.
There’s another way to remember this. This triangle is so pretty, let’s imagine it looks at itself in the mirror:
Reflected 30-60-90 triangle
So, now we have these two 60-degree angles on the bottom, and the 30-degree angle joins its reflected 30-degree angle to become another 60-degree angle. What is this?
It’s an equilateral triangle, with three equal angles and three equal sides.
So, in this triangle, if one side is 2, then all sides are 2. So, now what would our short leg be? It’s half of the bottom side, so it’d be 1.
Of course, you could also use your trigonometry wizardry. Or, if you know two sides, the Pythagorean Theorem. So, that’s like four ways to remember the 1:2:sqrt 3 rule.
Let’s see the 30-60-90 triangle in action. Okay, it’s not really going to play tennis or anything, but remembering the ratio can make these problems a breeze.
Here’s a 30-60-90 triangle where the short leg is 5:
The hypotenuse of this triangle is 10
What is the hypotenuse? We know that if the short leg is 1, the hypotenuse is 2. So, whatever we have for the short leg, the hypotenuse is just 2 times that. That means that here, it’ll be 10. Problem solved.
What about the long leg? That’s going to be 5 times the square root of 3. Sometimes, we’ll see 5 root 3 as an answer choice. If so, great! If not, we just need to do the math. The square root of 3 is about 1.73. And 5 times 1.73 is about 9.
Okay, here’s a tougher one:
The hypotenuse is about 13
In this triangle, the long leg is 11. What are the other two sides? Let’s start with the short side, which is x . We can just set up an equation using our ratio. We know it’s 1/(root 3), and that equals x /11. Cross multiply to get x (root 3) = 11. Then x = 11/(root 3). And that’s going to be about 6.
Now, the hypotenuse is y . So, we do y /11 = 2/(root 3). That gets us y (root 3) = 11 2, which is 22. So, y = 22/(root 3). That’ll be about 13.
Okay, so that was our Serena Williams - super talented, stylish, exciting. There’s another special right triangle that is a little more like Roger Federer - still super talented and exciting, though maybe a little less stylish. That’s our 45-45-90 triangle .
Watch what happens if you turn this triangle:
Now, do you notice what makes this triangle special? It’s an isosceles triangle! And it’s the only kind of isosceles right triangle you can make. Now, what’s great about isosceles triangles? The two legs are always the same. That makes finding the hypotenuse easy if you just know one of the legs - just use the Pythagorean Theorem.
But you can also remember this handy rule. The ratio of the sides is 1:1:sqrt 2. So, whatever one of the legs is, the hypotenuse is just that number times the square root of 2.
Also, note that if you put two of these triangles together, you get a square. Therefore, the area of one of these triangles is just 1/2 times the square of one of its legs.
Let’s practice finding the lengths of sides. Here’s a 45-45-90 triangle with a leg that is 7:
The hypotenuse is 10 for this triangle
What is the length of the hypotenuse? It’s just 7 times the square root of 2. That’s about 10. That’s it!
What about this one?
The length of the leg is about 12
We know the hypotenuse is 17. What is the length of the leg? That’s going to be 17/(root 2). And that’s going to be about 12.
In summary, we looked at two special types of right triangles:
The first is the 30-60-90 triangle . In this triangle, we call the side opposite the 30-degree angle the short leg and the side opposite the 60-degree angle the long leg. If the short leg is 1, the long leg is the square root of 3, and the hypotenuse is 2. In 30-60-90 triangles, the sides are always in this ratio.
Next, we looked at the 45-45-90 triangle . These are isosceles triangles. The two legs are always the same length. And the hypotenuse is just the leg times the square root of 2. And that’s game, set, match!
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