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Category: Mathematics
Special Right Triangles
Discover some special characteristics about 45°45°90° triangles!
A 45° – 45° – 90° triangle is an isosceles triangle… the two legs are the same length. The hypotenuse is always the length of the leg times the square root of two! So one quick and easy trick to help you remember is that there are two angles that are the same, so two sides are the same, and the third side will ALWAYS be the length of the leg multiplied by the square root of two!
The KEY is that you EXPECT to see the square root of two on the hypotenuse… so what happens if the hypotenuse is a whole number? Let’s say the hypotenuse is 14… what will the length of the legs be?
Remember, you EXPECTED to see the square root of two on the hypotenuse, but it isn’t there, so all you do is divide the hypotenuse by two, then multiply that figure by the square root of two and that is the length of the legs! (Remember, the two legs are the same length!) So in this case, each leg will be 7 times the square root of two!
Let’s give it a try… You have a 45°45°90° triangle and one of the legs is 6. What are the length of the other leg and the hypotenuse?
Did you get 6 for the other leg, and 6 times the square root of two for the hypotenuse?
Let’s try another one! You have a 45°45°90° triangle and the hypotenuse is 24. What are the lengths of the legs? (Hint: remember what you EXPECT to see!)
Did you get 12 times the square root of two for each leg?
What would the length of the legs be if the hypotenuse is 15?
All you have to do, is remember what you expect to see! If you got 7.5 times the square root of two, you are right on the money!!
Remember the KEY is what you EXPECT to see!! Click on the video below to see another explanation of this relationship.
Another special right triangle is the 30°60°90°.
The 30°60°90° has a very special relationship between it’s sides. First, you can tell that all three sides will be different lengths because the three angles are different. Similar to what we learned in the 45°45°90° triangle, the KEY is what you EXPECT to see!
With the 30°60°90° triangle, the first thing you do is locate the 30° angle. No matter where it is, the side opposite (or across) from the angle will be the shortest side of the triangle. That side is the KEY! From there, you can calculate the length of the other two sides of the triangle quickly and easily because the hypotenuse (the side opposite of the right (90°) angle will ALWAYS be twice as big as the short side!
The last side, the side opposite o the 60° angle is ALWAYS the short side times the square root of three! So there is a simple trick to help you remember which special triangle uses which square root… the 30°60°90° has three different angles so it has the square root of three, where the 45°45°90° triangle has two different angles so it uses the square root of two!
Let’s try a few problems! You have a 30°60°90° triangle and the side opposite the 30° angle is 6, what are the lengths of the other two sides?
If you said the side opposite the 90° angle is 12 and the side opposite the 60° angle is 6 times the square root of three you got it!!
What if the hypotenuse is 30? What would the other two sides be?
The short side (side opposite the 30° angle) is 15, and the side opposite the 60° angle is 15 times the square root of three!
Let’s see if you can figure this one out. (Hint: use the same skill set you learned with the 45°45°90° triangle and what you EXPECT to see!) You have a 30°60°90° triangle and the side opposite the 60° angle is 15. What are the lengths of the other two sides?
Did you get it? First, what did you expect to see on the side opposite the 60° angle? The square root of three right? But it isn’t there, so, in this case, you divide whatever value is there by three… in this case you get 5. So the short side (side opposite the 30° angle) will be 5 times the square root of three! The hypotenuse is twice as big, so it will be 10 times the square root of three!
If you practice these exercises until you remember what you EXPECT to SEE, you will have no problems being able to solve these special right triangles! Click below to hear a fun rap to help you remember the relationship of the 30°60°90° triangle.
Other Special Right Triangles
There are some other quick ways to solve for the lengths of the sides of special right triangles… these are your most common right triangles also known as:
Pythagorean Triples!

3 – 4 – 5

5 – 12 – 13

7 – 24 – 25

8 – 15 – 17
The great thing about knowing these triangles, is that any multiple will also be a right triangle! Let’s take a look…
So a triangle with sides that are 3″, 4″, and 5″ will ALWAYS be a right triangle. It could be 3 cm, 4 cm, 5 cm; 3′, 4′, 5′; 3 mi, 4 mi, 5 mi… in each case, they will each be a right triangle! The KEY is what you EXPECT to SEE! The longest side, is ALWAYS the hypotenuse. So, if I tell you the hypotenuse is 5 mm, and one of the other sides of the triangle is 4 mm and ask you, what is the length of the third side? 3 mm is right on the money! Remember, it works with any multiple as well! 345, 6810,91215, 121620, 152025, and so on and so on! If you know these four special right triangles, you will be able to solve a number of problems quickly and easily!
Click below to see more fun right triangle videos!
This next video is AMAZING!!! Another great resource to have up your sleeve!
Let’s do a quick review!
Let’s check for understanding!
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Awesome Math Facts
As you learned in your 8th grade math class and reviewed in your 9th grade Integrated I course, solving systems of linear equations by linear combination.
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