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Mia
11/29/2025
Geometry
Congruent Chords & Arcs
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Nov 29, 2025
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Understanding Congruent Chords and Arcs
Mia
@mialovesu16
1 / 4
Congruent Chords & Arcs
When working with circles, two chords are congruent (equal in length) if and only if their corresponding arcs have equal measures or they're the same distance from the center of the circle.
The key relationships to remember are:
- If AB = CD (congruent chords), then their arcs are equal
- If AB = CD, then they're equidistant from the center
- When a diameter or radius is perpendicular to a chord, it cuts both the chord and its arc exactly in half
Quick Tip: When solving chord problems, look for equal chords or arcs first - this relationship is your key to finding unknown values!
For example, in problem 1, we know that equal chords create equal arcs, so we can write 115 = 7x + 24, which gives us x = 13. Similarly in problem 3, equal arcs mean that 9x - 34 = 4x + 1, so x = 7.
When working with perpendicular lines, remember that they create right angles and bisect (cut in half) both chords and arcs. This principle helps you solve problems involving distances and arc measures.
Applying Chord & Arc Relationships
When solving circle problems, start by identifying what's equal - either chords, arcs, or distances from the center. This gives you equations you can solve.
For problem 7, we know that equidistant chords have equal lengths, so 11x + 15 = 32x - 27. Solving this gives x = 2, which means DE = 11(2) + 15 = 37. In problem 9, with RS = 18 and arc TY = 42°, we can find multiple measurements by applying chord-arc relationships.
When solving triangle problems in circles, use the Pythagorean theorem . In problem 13, we have a right triangle with sides 5 and x, and hypotenuse 13. Solving 5² + x² = 13² gives us x = 12, so VW = 24 (the full chord).
Remember: When a chord is given, the corresponding arc is twice the central angle. This relationship is super helpful for converting between chord and arc measurements!
For more complex problems, you might need to use inverse trigonometric functions. In problem 14, we find the arc measure using Sin⁻¹(12/13), which equals approximately 67.4°. Similarly, in problem 16, we use Cos⁻¹(8/17) to find an angle of 61.9°, making the arc measure 123.9°.
Chord & Arc Problem-Solving Techniques
Circles might seem complicated, but they follow predictable patterns. The key is recognizing when to use which relationship.
When solving equations with congruent chords or arcs, set up an equality and solve for the variable. In problem 1, we set up 59 = 10x - 31, giving us x = 9. For problem 2, the equation 7x - 39 = 87 yields x = 18.
Sometimes you'll need to work with multiple steps. In problem 3, we have 2 = 244, which simplifies to 26x - 42 = 244, giving us x = 11. The "2" at the beginning tells us we're dealing with a central angle that's twice the inscribed angle.
Problem-Solving Strategy: Always start by identifying what's equal in the problem (chords, arcs, or distances) and use that to write your equation!
When finding arc measures in a circle, remember that a full circle is 360°. In problem 7, after finding x = 14, we calculate mBAD = 2(14) - 53 = 73°. For problem 8, after solving 8x - 56 = 5x + 22 to get x = 26, we find mLP = 3(26) - 56 = 22°.
These techniques work for any chord and arc problem, so practice identifying the relationships and setting up equations.
Advanced Circle Relationships
Circle geometry becomes even more interesting when we combine multiple concepts. The power of these relationships lies in how they connect different parts of the circle.
In problem 9, when JG = JF (equidistant from center), we can determine that ED = 26 (twice GD) and find arc measures like mCD = 136° and mHD = 68°. This shows how one relationship can help us find multiple values.
For triangles in circles, the Pythagorean theorem remains your best friend. In problem 13, we use 9² + x² = 15² to find x = 12, making NK = 12. This lets us determine that JK = 24 in problem 15 (the entire chord).
Insight: When working with arcs and trigonometry, remember that you're finding angles in relation to the radius - this is why trig functions are so useful in circle problems!
Trigonometric functions help with finding arc measures when we know chord lengths. In problem 14, cos y = 4/15 gives us y = 53.1°, so the arc measure is 53.1°. For problem 16, we find mJPK = 253.8° (the major arc) by subtracting from 360°.
By combining these techniques - setting up equations for congruent chords, using the Pythagorean theorem, and applying trigonometric functions - you can tackle even the most challenging circle problems.
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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
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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!
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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
Understanding Congruent Chords and Arcs
Mia
@mialovesu16
Circle geometry is a powerful tool that helps us understand the relationships between chords, arcs, and angles in circles. In these lessons, you'll discover how congruent chords relate to arcs and distances from the center, and how to solve problems... Show more
Congruent Chords & Arcs
When working with circles, two chords are congruent (equal in length) if and only if their corresponding arcs have equal measures or they're the same distance from the center of the circle.
The key relationships to remember are:
- If AB = CD (congruent chords), then their arcs are equal
- If AB = CD, then they're equidistant from the center
- When a diameter or radius is perpendicular to a chord, it cuts both the chord and its arc exactly in half
Quick Tip: When solving chord problems, look for equal chords or arcs first - this relationship is your key to finding unknown values!
For example, in problem 1, we know that equal chords create equal arcs, so we can write 115 = 7x + 24, which gives us x = 13. Similarly in problem 3, equal arcs mean that 9x - 34 = 4x + 1, so x = 7.
When working with perpendicular lines, remember that they create right angles and bisect (cut in half) both chords and arcs. This principle helps you solve problems involving distances and arc measures.
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Applying Chord & Arc Relationships
When solving circle problems, start by identifying what's equal - either chords, arcs, or distances from the center. This gives you equations you can solve.
For problem 7, we know that equidistant chords have equal lengths, so 11x + 15 = 32x - 27. Solving this gives x = 2, which means DE = 11(2) + 15 = 37. In problem 9, with RS = 18 and arc TY = 42°, we can find multiple measurements by applying chord-arc relationships.
When solving triangle problems in circles, use the Pythagorean theorem . In problem 13, we have a right triangle with sides 5 and x, and hypotenuse 13. Solving 5² + x² = 13² gives us x = 12, so VW = 24 (the full chord).
Remember: When a chord is given, the corresponding arc is twice the central angle. This relationship is super helpful for converting between chord and arc measurements!
For more complex problems, you might need to use inverse trigonometric functions. In problem 14, we find the arc measure using Sin⁻¹(12/13), which equals approximately 67.4°. Similarly, in problem 16, we use Cos⁻¹(8/17) to find an angle of 61.9°, making the arc measure 123.9°.
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Chord & Arc Problem-Solving Techniques
Circles might seem complicated, but they follow predictable patterns. The key is recognizing when to use which relationship.
When solving equations with congruent chords or arcs, set up an equality and solve for the variable. In problem 1, we set up 59 = 10x - 31, giving us x = 9. For problem 2, the equation 7x - 39 = 87 yields x = 18.
Sometimes you'll need to work with multiple steps. In problem 3, we have 2 = 244, which simplifies to 26x - 42 = 244, giving us x = 11. The "2" at the beginning tells us we're dealing with a central angle that's twice the inscribed angle.
Problem-Solving Strategy: Always start by identifying what's equal in the problem (chords, arcs, or distances) and use that to write your equation!
When finding arc measures in a circle, remember that a full circle is 360°. In problem 7, after finding x = 14, we calculate mBAD = 2(14) - 53 = 73°. For problem 8, after solving 8x - 56 = 5x + 22 to get x = 26, we find mLP = 3(26) - 56 = 22°.
These techniques work for any chord and arc problem, so practice identifying the relationships and setting up equations.
Sign up to see the contentIt's free!
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Advanced Circle Relationships
Circle geometry becomes even more interesting when we combine multiple concepts. The power of these relationships lies in how they connect different parts of the circle.
In problem 9, when JG = JF (equidistant from center), we can determine that ED = 26 (twice GD) and find arc measures like mCD = 136° and mHD = 68°. This shows how one relationship can help us find multiple values.
For triangles in circles, the Pythagorean theorem remains your best friend. In problem 13, we use 9² + x² = 15² to find x = 12, making NK = 12. This lets us determine that JK = 24 in problem 15 (the entire chord).
Insight: When working with arcs and trigonometry, remember that you're finding angles in relation to the radius - this is why trig functions are so useful in circle problems!
Trigonometric functions help with finding arc measures when we know chord lengths. In problem 14, cos y = 4/15 gives us y = 53.1°, so the arc measure is 53.1°. For problem 16, we find mJPK = 253.8° (the major arc) by subtracting from 360°.
By combining these techniques - setting up equations for congruent chords, using the Pythagorean theorem, and applying trigonometric functions - you can tackle even the most challenging circle problems.
We thought you’d never ask...
What is the Knowunity AI companion?
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.
Where can I download the Knowunity app?
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Is Knowunity really free of charge?
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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