Hi Everyone!

On this page you will find some material about Lesson 23. Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Lesson 23: Special Triangles


In this section you will find some important information about the specific resources related to this lesson:

  • the learning outcomes,
  • the section in the textbook,
  • the WeBWorK homework sets,
  • a link to the pdf of the lesson notes,
  • a link to a video lesson.

Learning Outcomes. (from Coburn and Herdlick’s Trigonometry book)

  • Use the vocabulary associated with a study of angles and triangles.
  • Find and recognize coterminal angles.
  • Find fixed ratios of the sides of special triangles.

Topic. This lesson covers

Section 1.1: Special Triangles.

WeBWorK. There is one WeBWorK assignment on today’s material:


Lesson Notes.

These notes are used in Lessons 22-26. Today’s lesson is on pages 1-3.

Video Lesson.

Video Lesson 23 (based on Lesson 23 Notes)

This video is used in Lessons 22-26. For today’s lesson, watch from [1:44] to [10:46].



Warmup Questions

These are questions on fundamental concepts that you need to know before you can embark on this lesson. Don’t skip them! Take your time to do them, and check your answer by clicking on the “Show Answer” tab.

Warmup Question 1

What is the sum of the angles of a triangle?

Show Answer 1


Warmup Question 2

What is a right triangle?

Show Answer 2

It is a triangle that has an angle of 90^\circ, that is, a right angle.

Warmup Question 3

What are the sides of a right triangle called?

Show Answer 3

Hypotenuse and legs.

Warmup Question 4

In a right triangles, if one of the acute angles measures 30^\circ, what is the measure of the other angle?

Show Answer 4

The sum of the angles in a triangle is 180^{\circ}. In a right angle with an acute angle of 30^\circ, the third angle is


Warmup Question 5

Solve for x

    \[\dfrac{3}{x}=\dfrac{\sqrt 3}{7}.\]

Show Answer 5

    \[\dfrac{3}{x}=\dfrac{\sqrt 3}{7}\]

    \[\sqrt 3 x=21\]

    \[x=\dfrac{21}{\sqrt 3}\]

    \[x=\dfrac{21\sqrt 3}{\sqrt 3\sqrt 3}\]

    \[x=\dfrac{21\sqrt 3}{3}\]

    \[x=7\sqrt 3\]


If you are not comfortable with the Warmup Questions, don’t give up! Click on the indicated lesson for a quick catchup. A brief review will help you boost your confidence to start the new lesson, and that’s perfectly fine.

Need a review? Check Lesson ??

Quick Intro

This is like a mini-lesson with an overview of the main objects of study. It will often contain a list of key words, definitions and properties – all that is new in this lesson. We will use this opportunity to make connections with other concepts. It can be also used as a review of the lesson.

A Quick Intro to Special Triangles

Key Words. Similar triangles, special triangles, 30-60-90, 45-45-90, leg, hypotenuse.

There are two triangles that are called special, because their sides are in a special proportion.

\bigstar 30-60-90

This is a right triangle whose acute angles are 30^\circ and 60^\circ. When the hypotenuse is 2,

\bullet the side opposite to 30^\circ is the short leg and has length 1

\bullet the side opposite to 60^\circ is the long leg and has length \sqrt 3.

Any 30-60-90 triangle is similar to the above triangle. If we know one of the sides of a 30-60-90 triangle, one can use the ratios to find the two other sides.

\bigstar 45-45-90

This is a right triangle whose acute angles are both 45^\circ, so it is isosceles. When the legs measure 1, the hypotenuse measures \sqrt 2.

Any 45-45-90 triangle is similar to the above triangle. If we know one of the sides of a 45-45-90 triangle, one can use the ratios to find the two other sides.

Video Lesson

Many times the mini-lesson will not be enough for you to start working on the problems. You need to see someone explaining the material to you. In the video you will find a variety of examples, solved step-by-step – starting from a simple one to a more complex one. Feel free to play them as many times as you need. Pause, rewind, replay, stop… follow your pace!

Video Lesson

A video lesson on Special Triangles [32:45]

This video is used in Lessons 22-26. For today’s lesson, watch from [1:44] to [10:46].

In the video you will see

  • an example of a 45-45-90 triangle with sides measuring 4, 4 and 4\sqrt 2
  • an example of a 30-60-90 triangle with sides measuring 5, 5\sqrt 3 and 10
  • how to find the missing sides of a 30-60-90 triangle whose long leg measures 2
  • how to find the missing sides of a 45-45-90 triangle whose hypotenuse measures 3

Try Questions

Now that you have read the material and watched the video, it is your turn to put in practice what you have learned. We encourage you to try the Try Questions on your own. When you are done, click on the “Show answer” tab to see if you got the correct answer.

Try Question 1

Find x and y. Leave the answer in simplest radical form.

Show Answer 1

The triangle is a 45-45-90 triangle. It is isosceles, so both legs measure the same. This means that x=3. The hypotenuse is

    \[y=3\sqrt 2.\]

Try Question 2

Find x and y. Leave the answer in simplest radical form.

Show Answer 2

The triangle is a 30-60-90 triangle. The hypotenuse has length 8. The short leg is


The long leg is

    \[x=4\sqrt 3.\]


You should now be ready to start working on the WeBWorK problems. Doing the homework is an essential part of learning. It will help you practice the lesson and reinforce your knowledge.


It is time to do the homework on WeBWork:


When you are done, come back to this page for the Exit Questions.

Exit Questions

After doing the WeBWorK problems, come back to this page. The Exit Questions include vocabulary checking and conceptual questions. Knowing the vocabulary accurately is important for us to communicate. You will also find one last problem. All these questions will give you an idea as to whether or not you have mastered the material. Remember: the “Show Answer” tab is there for you to check your work!

Exit Questions

  • What makes the 45-45-90 and the 30-60-90 triangles special? 
  • If the hypotenuse of the the 30-60-90 has length 2, what is the length of the side opposite 60? 
  • How many ratios are there?  And what are they?  How can they be written in terms of words like opposite, adjacent, and hypotenuse and must these terms be in relation to an angle?

\bigstar Find x and y. Leave the answer in simplest radical form.

Show Answer

The triangle is a 30-69-90 triangle. The long leg has length 12. The short leg is

    \[x=\dfrac{12}{\sqrt 3}=\dfrac{12\sqrt 3}{\sqrt 3\sqrt 3}=\dfrac{12\sqrt 3}{3}=4\sqrt 3.\]

The hypotenuse is twice the short leg

    \[y=2\cdot 4\sqrt 3=8\sqrt 3.\]

Need more help?

Don’t wait too long to do the following.

  • Watch the additional video resources.
Additional video resources on Special Triangles
  • Talk to your instructor.
  • Form a study group.
  • Visit a tutor. For more information, check the tutoring page.