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Geometry
Dec 11, 2025
208
3 pages
Ashley Ngokwere @d1g1t4l
Geometry gets exciting when we explore how segments relate to each other in circles! These relationships give us... Show more
When two chords cross each other inside a circle, something amazing happens. The product of the segments on one chord equals the product of the segments on the other chord. This pattern works every time!
To use this theorem, simply multiply the segments on each chord and set them equal. For example, if chord AB intersects chord CD, then (segment A) × (segment B) = (segment C) × (segment D).
Let's see how this works If one chord has segments of 14 and 5, and it intersects another chord with segments of 7 and what we're solving for (x), we write 14 × 5 = 7 × x, giving us x = 10.
💡 Quick Tip When solving these problems, always identify the four segments created by the intersection point, then set up your equation using the product relationship.
When two lines (secants) pass through a point outside a circle and intersect the circle, another pattern emerges. The product of the entire first secant and its external part equals the product of the entire second secant and its external part.
This gives us the formula B = D, where the letters represent the different segment lengths. When you're given three of these values, you can solve for the fourth!
What happens when a tangent line (which touches the circle at exactly one point) meets a secant line outside the circle? The square of the tangent length equals the product of the secant's external part and its entire length.
This relationship is written as Tangent² = External secant × Whole secant length. For example, if a tangent has length x, and a secant has an external part of 5 and total length of 20, we'd write x² = 5 × 20, giving us x = 10.
This pattern is super useful for solving circle problems when you have a mix of tangents and secants!
🔑 Remember The tangent-secant formula uses the square of the tangent, not just the tangent itself. This is different from the secant-secant relationship.
Circles have beautiful symmetry! In a circle, congruent central angles create congruent chords. Similarly, congruent chords create congruent arcs, and congruent arcs create congruent central angles.
These relationships work like a cycle, each implying the others. When you see congruent parts in a circle, you can immediately identify other parts that must also be congruent.
When a radius or diameter is perpendicular to a chord, it creates a special relationship - it bisects (cuts in half) both the chord and its corresponding arc.
This means if radius AB is perpendicular to chord CD, then chord CD is divided into two equal parts, and arc CD is also divided into two equal parts.
This property gives us an excellent shortcut for finding chord lengths using the Pythagorean theorem. Since we form a right triangle, we can use a² + b² = c² to find missing lengths.
📝 Important When using the Pythagorean theorem with circles, identify the right triangle first, then apply the formula to find the unknown sides.
For example, if a radius of length 9 is perpendicular to a chord, and the distance from the center to the chord is 6, we can find half the chord length using 6² + x² = 9². This gives us x = √45 = 3√5, so the full chord length is 6√5 or approximately 13.4.
Mastering these circle segment relationships will help you tackle even the most challenging geometry problems with confidence!
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.
7
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Explore the fundamental circle theorems, including the alternate segment theorem, properties of cyclic quadrilaterals, and relationships between angles in circles. This summary covers key concepts such as inscribed angles, central angles, and tangent properties, providing a clear understanding for students preparing for exams.
Maths***there may be grammatical or spelling errors and many abbreviations as these were originally just for me; it is not guaranteed to be 100% accurate***
GeometryGeometry Unit 10 lesson; Tangents
GeometrySegment relationships
GeometryIncludes segments of chords theorem, segments of secants theorem and segments of secants and tangent theorem.
GeometryThis page talks about the two secant segments theorem, the tangent secant segment theorem, and how to solve problems for each.
GeometryApp 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
Ashley Ngokwere
@d1g1t4l
Geometry gets exciting when we explore how segments relate to each other in circles! These relationships give us powerful tools to find missing values using simple equations, no matter how complex the circle diagram looks.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When two chords cross each other inside a circle, something amazing happens. The product of the segments on one chord equals the product of the segments on the other chord. This pattern works every time!
To use this theorem, simply multiply the segments on each chord and set them equal. For example, if chord AB intersects chord CD, then (segment A) × (segment B) = (segment C) × (segment D).
Let's see how this works: If one chord has segments of 14 and 5, and it intersects another chord with segments of 7 and what we're solving for (x), we write 14 × 5 = 7 × x, giving us x = 10.
💡 Quick Tip: When solving these problems, always identify the four segments created by the intersection point, then set up your equation using the product relationship.
When two lines (secants) pass through a point outside a circle and intersect the circle, another pattern emerges. The product of the entire first secant and its external part equals the product of the entire second secant and its external part.
This gives us the formula B = D, where the letters represent the different segment lengths. When you're given three of these values, you can solve for the fourth!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
What happens when a tangent line (which touches the circle at exactly one point) meets a secant line outside the circle? The square of the tangent length equals the product of the secant's external part and its entire length.
This relationship is written as: Tangent² = External secant × Whole secant length. For example, if a tangent has length x, and a secant has an external part of 5 and total length of 20, we'd write x² = 5 × 20, giving us x = 10.
This pattern is super useful for solving circle problems when you have a mix of tangents and secants!
🔑 Remember: The tangent-secant formula uses the square of the tangent, not just the tangent itself. This is different from the secant-secant relationship.
Circles have beautiful symmetry! In a circle, congruent central angles create congruent chords. Similarly, congruent chords create congruent arcs, and congruent arcs create congruent central angles.
These relationships work like a cycle, each implying the others. When you see congruent parts in a circle, you can immediately identify other parts that must also be congruent.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When a radius or diameter is perpendicular to a chord, it creates a special relationship - it bisects (cuts in half) both the chord and its corresponding arc.
This means if radius AB is perpendicular to chord CD, then chord CD is divided into two equal parts, and arc CD is also divided into two equal parts.
This property gives us an excellent shortcut for finding chord lengths using the Pythagorean theorem. Since we form a right triangle, we can use a² + b² = c² to find missing lengths.
📝 Important: When using the Pythagorean theorem with circles, identify the right triangle first, then apply the formula to find the unknown sides.
For example, if a radius of length 9 is perpendicular to a chord, and the distance from the center to the chord is 6, we can find half the chord length using: 6² + x² = 9². This gives us x = √45 = 3√5, so the full chord length is 6√5 or approximately 13.4.
Mastering these circle segment relationships will help you tackle even the most challenging geometry problems with confidence!
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.
7
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Explore the fundamental circle theorems, including the alternate segment theorem, properties of cyclic quadrilaterals, and relationships between angles in circles. This summary covers key concepts such as inscribed angles, central angles, and tangent properties, providing a clear understanding for students preparing for exams.
Maths***there may be grammatical or spelling errors and many abbreviations as these were originally just for me; it is not guaranteed to be 100% accurate***
GeometryGeometry Unit 10 lesson; Tangents
GeometrySegment relationships
GeometryIncludes segments of chords theorem, segments of secants theorem and segments of secants and tangent theorem.
GeometryThis page talks about the two secant segments theorem, the tangent secant segment theorem, and how to solve problems for each.
GeometryApp 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
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