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Special Right Triangles (Lesson 9.4)

Learning Targets
  • Use properties of 30-60-90 triangles to find side lengths.
  • Use properties of 45-45-90 triangles to find side lengths.
Activity: What Makes These Triangles Special?

Experience First

In today’s lesson, we begin the transition from right triangle trig to the trigonometry with the unit circle. One of the main goals in this unit is a deep understanding of the unit circle. This will rely heavily on the use of special right triangles. In this lesson, students will investigate patterns in 30-60-90 and 45-45-90 triangles. In the next lesson we will establish conventions for angles in the coordinate plane. Then finally we will put it all together to create a unit circle in Lesson 9.6.


Students should work in groups to complete the entire first page. As you are walking around working with groups, make sure they are correctly reducing their answers into reduced radical form. If they are not, point out someone in their group who has done it correctly and say, “Oh, I notice you two have different answers. Can you talk about that?”

Prior Knowledge

  • Students need to be able to write numbers in reduced radical form. If you think they haven’t learned this before or if they need a refresher, it would be worth taking the first 5-10 minutes of class to go over this.

  • All other prior knowledge was covered in previous lessons. See Lesson 9.1 for more.

Guiding Questions

  • If we know two sides of a right triangle, how can we find the third side?

  • What do you notice about the angles in these triangles?

  • Say to a student who has reduced correctly, “When I did the Pythagorean Theorem, I got the square root of 52. How did you get your answer?”

  • What patterns are you noticing in the table? What if I had a triangle like this that had a short side of 11? What would the other sides be?

Formalize Later

We want to be careful here with how we label each side of the triangle. We prefer legs (short and long) and hypotenuse. You could label differently, like medium side for example, but whatever you do, just make sure you’re consistent. Your labels in the margins should match the labels in the QuickNotes.


When students work on the Check Your Understanding, they will have the hardest time when they have to divide by the radical. Ideally, the final answers will not have a radical in the denominator. If students aren’t sure how to do this, spending a little time going over it now will help when we get to the Unit Circle.

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