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## High school geometry

### Course: High school geometry>Unit 5

Lesson 3: Special right triangles

# Special right triangles review

Learn shortcut ratios for the side lengths of two common right triangles: 45°-45°-90° and 30°-60°-90° triangles. The ratios come straight from the Pythagorean theorem.

## 30-60-90 triangles

30-60-90 triangles are right triangles whose acute angles are 30, degrees and 60, degrees. The sides in such triangles have special proportions:
A thirty-sixty-ninety triangle. The length of the shorter leg of the triangle is one half h units. The length of the longer leg of the triangle is square root three over two times h. The length of the hypotenuse of the triangle is h units.

## 45-45-90 triangles

45-45-90 triangles are right triangles whose acute angles are both 45, degrees. This makes them isosceles triangles, and their sides have special proportions:
A forty-five-forty-five-ninety triangle. The length of both legs are k units. The length of the hypotenuse of the triangle is square root of two times k units.
The special properties of both of these special right triangles are a result of the Pythagorean theorem.

Problem 1
• Current
A right triangle A B C has angle A being thirty degrees. Side A B is x units. Side B C is six units.
A, B, equals

Want to practice more problems like this? Check out this exercise.

## Want to join the conversation?

• im so used to doing a2+b2=c 2 what has changed I do not understand
• With 45-45-90 and 30-60-90 triangles you can figure out all the sides of the triangle by using only one side. If you know one short side of a 45-45-90 triangle the short side is the same length and the hypotenuse is root 2 times larger. If you know the hypotenuse of a 45-45-90 triangle the other sides are root 2 times smaller. If you know the 30-degree side of a 30-60-90 triangle the 60-degree side is root 3 times larger and the hypotenuse is twice as long. if you know the 60-degree side of a 30-60-90 triangle the 30-degree side is root 3 times smaller and the hypotenuse is 2/root 3 times longer. If you know the hypotenuse of a 30-60-90 triangle the 30-degree is half as long and the 60-degree side is root 3/2 times as long.
• What is the difference between congruent triangles and similar triangles?
• Congruent are same size and same shape
Similar are same shape but different size
Both have to have one to one correspondence between their angles, but congruent also has one to one correspondence between their sides, but similar sides are equally proportional
• what can i do to not get confused with what im doing ?
• I'd make sure I knew the basic skills for the topic. For special triangles some skills you need to master are: Angles, Square roots, and most importantly The Pythagorean Theorem. Another source you can use is the hints in the exercises, they can help guide you.
• How can you tell if a triangle is a 30 60 90 triangle vs a 45 45 90 triangle? Help!
• A 45 45 90 triangle is isosceles. The two legs are equal. A 30 60 90 triangle has the hypotenuse 2 times as long as the short leg. Hope this helps!
• Let's say that there is a 30-60-90 triangle and I need to figure out the side opposite of the 60 degree angle and the hypotenuse is something like 6 times the square root of 3. I know that to get the answer I need to multiply this by the square root of 3 over 2. Do I multiply everything or is there a certain time when I divide or do something with square roots and/or roots? Would the answer to this problem be 36 (square root of 3 times the square root of 3 to get 3, 2 times 6 to get 12, and 12 times 3 to get 36)?
• The answer to your problem is actually 9. You are correct about multiplying the square root of 3 / 2 by the hypotenuse (6 * root of 3), but your answer is incorrect. This is because if you multiply the square root of 3 by 6 times the root of three, that would be the same as multiplying 3 by 6 (because the square root of 3 squared is 3). 3 by 6 is 18, and that divided by 2 would equal 9, which is the correct answeer.
• if I get 30.1 degrees, is it still a special triangle
• No, but it is approximately a special triangle. I do not know how you can tell the difference on a protractor between 30 and 30.1 degrees.
• Can't you just use SOH CAH TOA to find al of these?
• Yes, but special right triangles have constant ratios, so if you learn how to do this, you can get answers faster.
• I use this trick on 30, 60, 90 triangles and I've never gotten a single wrong -
1. The small leg to the hypotenuse is times 2, Hypotenuse to the small leg is divided by 2.
2. The small leg (x) to the longer leg is x radical three

For Example-
Pretend that the short leg is 4 and we will represent that as "x." And we are trying to find the length of the hypotenuse side and the long side. To find the lengths of the hypotenuse from the short leg (x), all we have to do is x times 2, which in this case is 4 times 2. Four times 2 is 8. The length of the hypotenuse side is 8. That is how to find the hypotenuse from the short leg. But are we done yet? No, we are not. We still have to find the length of the long leg. Since the short leg (x) is 4, we have to do "x" radical three. I came to a conclusion that the long leg is 4 radical 3.

-This works everytime
• wow, thanks :)
• Are special right triangles still classified as right triangles?