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Right Triangles & Trigonometry

Special Right Triangles

534

•

Jun 27, 2023

•

10 pages

Easy Ways to Find Missing Sides in Special Right Triangles

user profile picture

C

@cnnotes

A comprehensive guide to understanding 45 45 90 triangle theorem... Show more

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

45-45-90 Triangle Theorem

This page explains the fundamental properties of the 45-45-90 triangle, which is a right triangle with two 45-degree angles.

Highlight: In a 45-45-90 triangle, both legs are equal in length, and the hypotenuse is √2 times the length of a leg.

Example: If a leg is length S, then the other leg is also S, and the hypotenuse is S√2.

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

30-60-90 Triangle Theorem

This page details the characteristics of the 30-60-90 triangle and its side length relationships.

Definition: A 30-60-90 triangle is a right triangle with angles of 30, 60, and 90 degrees.

Highlight: The hypotenuse is always twice the length of the shorter leg, while the longer leg is √3 times the shorter leg.

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Important Note on Applications

This page emphasizes the significance of memorizing these special triangles.

Quote: "These two triangles are very important to memorize. We will use them all semester and they are frequently used in other math classes."

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Examples Introduction

This page serves as a transition to practical applications of the theorems.

Vocabulary: Examples demonstrate how to apply theoretical knowledge to solve real mathematical problems.

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Example 1: 30-60-90 Triangle

This page demonstrates how to find missing sides in special right triangles when given the hypotenuse of a 30-60-90 triangle.

Example: Given a hypotenuse of 4 units, the short side oppositeto30°opposite to 30° is 2 units, and the long side oppositeto60°opposite to 60° is 2√3 units.

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Example 2: 45-45-90 Triangle

This page shows how to solve for missing sides in a 45-45-90 triangle when given one leg.

Example: When one leg is 9 units, the other leg is also 9 units, and the hypotenuse is 9√2 units.

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Example 3: Part 1

This page begins a multi-step example solving a 30-60-90 triangle with a known long side of 18 units.

Highlight: The solution process begins by using the relationship s√3 = 18 to find the value of s.

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Example 3: Part 2

This page continues the solution process, focusing on rationalizing denominators.

Example: The short side xx is calculated to be 6√3 units through rationalization of the denominator.

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Example 3: Part 3

This page concludes the example by finding the hypotenuse.

Highlight: The hypotenuse yy is calculated as 12√3 units by doubling the short side length.

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Special Right Triangles Introduction

This page introduces the concept of special right triangles, which are essential geometric figures with unique properties that make calculating side lengths more efficient.

Definition: Special right triangles are right triangles with specific angle measurements that create consistent relationships between their sides.



Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

Subjects

Geometry

Right Triangles & Trigonometry

Special Right Triangles

 

Geometry

•

534

•

Jun 27, 2023

•

10 pages

Easy Ways to Find Missing Sides in Special Right Triangles

user profile picture

C

@cnnotes

A comprehensive guide to understanding 45 45 90 triangle theorem and solving 30 60 90 triangle side lengths. This mathematical concept focuses on special right triangles and their unique properties for calculating missing sides.

  • Special right triangles (45-45-90 and... Show more

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

45-45-90 Triangle Theorem

This page explains the fundamental properties of the 45-45-90 triangle, which is a right triangle with two 45-degree angles.

Highlight: In a 45-45-90 triangle, both legs are equal in length, and the hypotenuse is √2 times the length of a leg.

Example: If a leg is length S, then the other leg is also S, and the hypotenuse is S√2.

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

30-60-90 Triangle Theorem

This page details the characteristics of the 30-60-90 triangle and its side length relationships.

Definition: A 30-60-90 triangle is a right triangle with angles of 30, 60, and 90 degrees.

Highlight: The hypotenuse is always twice the length of the shorter leg, while the longer leg is √3 times the shorter leg.

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Important Note on Applications

This page emphasizes the significance of memorizing these special triangles.

Quote: "These two triangles are very important to memorize. We will use them all semester and they are frequently used in other math classes."

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Examples Introduction

This page serves as a transition to practical applications of the theorems.

Vocabulary: Examples demonstrate how to apply theoretical knowledge to solve real mathematical problems.

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Example 1: 30-60-90 Triangle

This page demonstrates how to find missing sides in special right triangles when given the hypotenuse of a 30-60-90 triangle.

Example: Given a hypotenuse of 4 units, the short side oppositeto30°opposite to 30° is 2 units, and the long side oppositeto60°opposite to 60° is 2√3 units.

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Example 2: 45-45-90 Triangle

This page shows how to solve for missing sides in a 45-45-90 triangle when given one leg.

Example: When one leg is 9 units, the other leg is also 9 units, and the hypotenuse is 9√2 units.

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Example 3: Part 1

This page begins a multi-step example solving a 30-60-90 triangle with a known long side of 18 units.

Highlight: The solution process begins by using the relationship s√3 = 18 to find the value of s.

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Example 3: Part 2

This page continues the solution process, focusing on rationalizing denominators.

Example: The short side xx is calculated to be 6√3 units through rationalization of the denominator.

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Example 3: Part 3

This page concludes the example by finding the hypotenuse.

Highlight: The hypotenuse yy is calculated as 12√3 units by doubling the short side length.

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle) In a 45° - 45°-90° triangle, both legs are congruent and the length of the hypoten

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Special Right Triangles Introduction

This page introduces the concept of special right triangles, which are essential geometric figures with unique properties that make calculating side lengths more efficient.

Definition: Special right triangles are right triangles with specific angle measurements that create consistent relationships between their sides.

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user