Subjects
Biology
Chemistry
AP Chemistry
English
AP English Literature
Geometry
Dec 7, 2025
69
7 pages
Rylan Parady @rylanparady
Circles are all about special points and lines that follow specific rules. In this topic, you'll learn how... Show more
A circle is a set of points equidistant from a center point. When you see "circle P," that means P is the center.
There are four key segments and lines that interact with circles
Quick Tip Think of tangent lines as "just barely touching" the circle, while secants "cut through" the circle completely.
When two circles exist in the same plane, they can be
Circles can also share common tangent lines—either external tangents (not passing between the circles) or internal tangents (passing between them).
The relationship between tangent lines and radii is special and extremely useful. The Tangent Line to Circle Theorem states that a line is tangent to a circle if and only if it's perpendicular to the radius at the point of tangency.
This makes perfect sense when you think about it the radius points "straight out" from the circle, and the tangent line runs "straight across" at that point.
Another important property is the External Tangent Congruence Theorem. When you draw two tangent segments from a point outside a circle, those segments are always equal in length.
Remember Whenever you see a line tangent to a circle, there must be a radius perpendicular to it at the point of contact. This relationship is guaranteed!
These properties help us solve many geometry problems involving circles and tangent lines, especially when we need to find missing angles or lengths.
Chords have several important properties that help us solve circle problems
The Congruent Corresponding Chord Theorem tells us that in the same circle (or in congruent circles), two minor arcs are congruent if and only if their corresponding chords are congruent. Simply put equal chords create equal arcs, and vice versa.
When a diameter is perpendicular to a chord, something special happens. The Perpendicular Chord Bisector Theorem states that the diameter will bisect (cut in half) both the chord and the arc it creates. The converse is also true if one chord perpendicularly bisects another, then the first chord must be a diameter.
Make this practical When you need to find the exact center of a circle, draw two chords and find their perpendicular bisectors. The point where they intersect is the center!
The Equidistant Chord Theorem gives us another useful relationship two chords are congruent if and only if they're equidistant from the center. This means equal chords are always the same distance from the center of the circle.
When angles and polygons are inscribed in a circle, they follow predictable patterns.
The Inscribed Right Triangle Theorem gives us a way to identify right angles if a right triangle is inscribed in a circle, its hypotenuse must be a diameter. The reverse is also true—if one side of an inscribed triangle is a diameter, the angle opposite it must be 90°.
Quadrilaterals can be inscribed in circles too. The Inscribed Quadrilateral Theorem tells us that a quadrilateral can fit perfectly on a circle if and only if its opposite angles are supplementary (add up to 180°).
Test Tip On exams, look for problems with inscribed shapes. If you spot a quadrilateral inside a circle, immediately check if opposite angles add to 180°!
You can use these properties to solve for missing angle measures. For example, if you know a triangle is inscribed in a circle with one side as a diameter, you can immediately identify the opposite angle as 90°, which makes solving for other angles much easier.
Circles can be defined using coordinate geometry with some straightforward equations.
The simplest case is a circle centered at the origin (0,0). For any point (x,y) on this circle, the equation is x² + y² = r², where r is the radius. This comes directly from the Pythagorean theorem, since the distance from any point to the origin must equal the radius.
For circles centered at any point (h,k), the equation becomes ² + ² = r². This is called the standard equation of a circle. The and terms represent the horizontal and vertical distances from any point (x,y) on the circle to the center.
Pro Strategy When solving circle problems in the coordinate plane, first identify the center and radius, then plug them into the standard equation.
Finding the equation is straightforward if you know the center and radius. If you know the center and a point on the circle, you can find the radius by calculating the distance between these points, then form the equation.
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
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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
Rylan Parady
@rylanparady
Circles are all about special points and lines that follow specific rules. In this topic, you'll learn how circles relate to different types of lines and segments, and how to use these relationships to solve problems. These concepts are foundational... Show more
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Join milions of students
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A circle is a set of points equidistant from a center point. When you see "circle P," that means P is the center.
There are four key segments and lines that interact with circles:
Quick Tip: Think of tangent lines as "just barely touching" the circle, while secants "cut through" the circle completely.
When two circles exist in the same plane, they can be:
Circles can also share common tangent lines—either external tangents (not passing between the circles) or internal tangents (passing between them).
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The relationship between tangent lines and radii is special and extremely useful. The Tangent Line to Circle Theorem states that a line is tangent to a circle if and only if it's perpendicular to the radius at the point of tangency.
This makes perfect sense when you think about it: the radius points "straight out" from the circle, and the tangent line runs "straight across" at that point.
Another important property is the External Tangent Congruence Theorem. When you draw two tangent segments from a point outside a circle, those segments are always equal in length.
Remember: Whenever you see a line tangent to a circle, there must be a radius perpendicular to it at the point of contact. This relationship is guaranteed!
These properties help us solve many geometry problems involving circles and tangent lines, especially when we need to find missing angles or lengths.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Chords have several important properties that help us solve circle problems:
The Congruent Corresponding Chord Theorem tells us that in the same circle (or in congruent circles), two minor arcs are congruent if and only if their corresponding chords are congruent. Simply put: equal chords create equal arcs, and vice versa.
When a diameter is perpendicular to a chord, something special happens. The Perpendicular Chord Bisector Theorem states that the diameter will bisect (cut in half) both the chord and the arc it creates. The converse is also true: if one chord perpendicularly bisects another, then the first chord must be a diameter.
Make this practical: When you need to find the exact center of a circle, draw two chords and find their perpendicular bisectors. The point where they intersect is the center!
The Equidistant Chord Theorem gives us another useful relationship: two chords are congruent if and only if they're equidistant from the center. This means equal chords are always the same distance from the center of the circle.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When angles and polygons are inscribed in a circle, they follow predictable patterns.
The Inscribed Right Triangle Theorem gives us a way to identify right angles: if a right triangle is inscribed in a circle, its hypotenuse must be a diameter. The reverse is also true—if one side of an inscribed triangle is a diameter, the angle opposite it must be 90°.
Quadrilaterals can be inscribed in circles too. The Inscribed Quadrilateral Theorem tells us that a quadrilateral can fit perfectly on a circle if and only if its opposite angles are supplementary (add up to 180°).
Test Tip: On exams, look for problems with inscribed shapes. If you spot a quadrilateral inside a circle, immediately check if opposite angles add to 180°!
You can use these properties to solve for missing angle measures. For example, if you know a triangle is inscribed in a circle with one side as a diameter, you can immediately identify the opposite angle as 90°, which makes solving for other angles much easier.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Circles can be defined using coordinate geometry with some straightforward equations.
The simplest case is a circle centered at the origin (0,0). For any point (x,y) on this circle, the equation is x² + y² = r², where r is the radius. This comes directly from the Pythagorean theorem, since the distance from any point to the origin must equal the radius.
For circles centered at any point (h,k), the equation becomes ² + ² = r². This is called the standard equation of a circle. The and terms represent the horizontal and vertical distances from any point (x,y) on the circle to the center.
Pro Strategy: When solving circle problems in the coordinate plane, first identify the center and radius, then plug them into the standard equation.
Finding the equation is straightforward if you know the center and radius. If you know the center and a point on the circle, you can find the radius by calculating the distance between these points, then form the equation.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
2
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
App Store
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
Calculus 1
Algebra 1
Overige