# Special Right Triangle Discovery Activity

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I created this special right triangle discovery activity to help my trigonometry students discover (or rediscover) the patterns in the sides of the 30-60-90 and 45-45-90 special right triangles.

## 45-45-90 Special Right Triangle Discovery Activity

The page I’m showing you FIRST is actually what we did LAST.  Yes, I’m that teacher who makes my students discover the pattern for themselves.  Here’s the summary we wrote as a class AFTER we discovered the relationship that holds true for all 45-45-90 right triangles.

Here’s the first thing my students saw.  Yes, it’s a page with 12 squares that are cut by a diagonal.

Students had to determine the angle measures and use the Pythagorean Theorem to solve for the diagonal.  They continued to keep chugging along with the Pythagorean Theorem until someone in the class noticed a pattern.

After about the first four problems, students started to notice something peculiar.  Soon, they were racing through each question.  No Pythagorean Theorem needed.

We really did have to stop and think about how to do the problem when we were given the diagonal of the square instead of the length of the sides.  But, we persevered and figured it out!

To give my students some practice with 45-45-90 special right triangles, we did the One Big Square Task.

## 30-60-90 Special Right Triangle Discovery Activity

Finally, I wanted students to discover the pattern for 30-60-90 triangles the same way we discovered it for 45-45-90 triangles.  Here’s the summary sheet we filled out AFTER discovering the pattern ourselves:

Here are the problems I posed for my students in order to help them see the relationship for 30-60-90 right triangles.

For each equilateral triangle, students had to use the Pythagorean Theorem to solve for the altitude of the triangle.  We did this over and over and over until the pattern began to emerge.  Students were on a lookout for a pattern from the get-go because of their experience with the 45-45-90 activity.

You’re probably not surprised that I made a poof book for my students to practice special right triangles.  I’m kinda obsessed with these mini-books!

We also tested our new special right triangle knowledge with a Dragon Puzzle.