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13 Dec 2025
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Understanding Straight Lines: Simple Notes and Examples
hana
@hanajess777
1 / 4
Collinear Lines and Gradients
Collinear points all lie on the same straight line, and proving this is simpler than you might think. When points are collinear, all the line segments between them have identical gradients - that's your key to solving these problems.
To prove collinearity, calculate the gradients between different pairs of points using . If they're equal, state that the segments are parallel and identify the common point they share.
Here's a handy tip: positive gradients slope upwards (like climbing a hill), whilst negative gradients slope downwards. The gradient actually equals tan θ, where θ is the angle between the line and the x-axis.
Quick Check: For negative slopes, find the angle using tan⁻¹, then subtract from 180° to get the actual angle with the x-axis.
Distance Formula and Perpendicular Lines
Finding the distance between two points is like using Pythagoras' theorem on a coordinate plane. The formula might look intimidating, but it's just creating a right-angled triangle and finding the hypotenuse.
Perpendicular gradients follow a simple rule: . To find a perpendicular gradient, flip the original fraction upside down and change the sign. If your gradient is , the perpendicular gradient becomes .
When solving perpendicular line problems, rearrange the given equation into form first, then apply the perpendicular rule. Always leave distance answers in exact form like $\sqrt{5}$ unless specifically asked to round.
Pro Tip: Remember that perpendicular lines meet at 90°, so their gradients are negative reciprocals of each other.
Medians and Altitudes of Triangles
A median connects the midpoint of one side to the opposite vertex - think of it as a line from the middle of a triangle's side to the far corner. Start by finding the midpoint using , then calculate the gradient to the opposite vertex.
An altitude is quite different - it drops perpendicularly from a vertex to the opposite side, creating a right angle. This means you'll need to find the gradient of the opposite side first, then use the perpendicular gradient rule.
Both problems follow the same final step: substitute your gradient and coordinates into to find the equation. The tricky bit is identifying which gradient and points to use for each type of line.
Memory Aid: Medians go to midpoints, altitudes are always perpendicular to the opposite side.
Perpendicular Bisectors
A perpendicular bisector combines everything you've learned - it passes through the midpoint of a line segment and meets it at a right angle. You're essentially creating both a median and an altitude in one go.
Follow this systematic approach: find the midpoint of the original line segment, calculate the gradient of that segment, then invert and change the sign to get your perpendicular gradient. Finally, use the equation with your midpoint coordinates.
These problems often appear in exam questions because they test multiple skills simultaneously. Take your time with the arithmetic - small errors in fraction calculations can throw off your entire answer.
Exam Strategy: Always double-check that your perpendicular gradient multiplies with the original gradient to give -1.
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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
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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
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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
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Best app on earth! no words because it’s too good
Thomas R
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Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Understanding Straight Lines: Simple Notes and Examples
hana
@hanajess777
Ever wondered how to prove three points lie on a straight line or find the exact distance between two coordinates? These coordinate geometry skills are essential for your Higher Maths exam and surprisingly useful in real life applications like computer... Show more
Collinear Lines and Gradients
Collinear points all lie on the same straight line, and proving this is simpler than you might think. When points are collinear, all the line segments between them have identical gradients - that's your key to solving these problems.
To prove collinearity, calculate the gradients between different pairs of points using . If they're equal, state that the segments are parallel and identify the common point they share.
Here's a handy tip: positive gradients slope upwards (like climbing a hill), whilst negative gradients slope downwards. The gradient actually equals tan θ, where θ is the angle between the line and the x-axis.
Quick Check: For negative slopes, find the angle using tan⁻¹, then subtract from 180° to get the actual angle with the x-axis.
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Join milions of students
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Distance Formula and Perpendicular Lines
Finding the distance between two points is like using Pythagoras' theorem on a coordinate plane. The formula might look intimidating, but it's just creating a right-angled triangle and finding the hypotenuse.
Perpendicular gradients follow a simple rule: . To find a perpendicular gradient, flip the original fraction upside down and change the sign. If your gradient is , the perpendicular gradient becomes .
When solving perpendicular line problems, rearrange the given equation into form first, then apply the perpendicular rule. Always leave distance answers in exact form like $\sqrt{5}$ unless specifically asked to round.
Pro Tip: Remember that perpendicular lines meet at 90°, so their gradients are negative reciprocals of each other.
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
Medians and Altitudes of Triangles
A median connects the midpoint of one side to the opposite vertex - think of it as a line from the middle of a triangle's side to the far corner. Start by finding the midpoint using , then calculate the gradient to the opposite vertex.
An altitude is quite different - it drops perpendicularly from a vertex to the opposite side, creating a right angle. This means you'll need to find the gradient of the opposite side first, then use the perpendicular gradient rule.
Both problems follow the same final step: substitute your gradient and coordinates into to find the equation. The tricky bit is identifying which gradient and points to use for each type of line.
Memory Aid: Medians go to midpoints, altitudes are always perpendicular to the opposite side.
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
Perpendicular Bisectors
A perpendicular bisector combines everything you've learned - it passes through the midpoint of a line segment and meets it at a right angle. You're essentially creating both a median and an altitude in one go.
Follow this systematic approach: find the midpoint of the original line segment, calculate the gradient of that segment, then invert and change the sign to get your perpendicular gradient. Finally, use the equation with your midpoint coordinates.
These problems often appear in exam questions because they test multiple skills simultaneously. Take your time with the arithmetic - small errors in fraction calculations can throw off your entire answer.
Exam Strategy: Always double-check that your perpendicular gradient multiplies with the original gradient to give -1.
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
You can download the app from Google Play Store and Apple App Store.
Is Knowunity really free of charge?
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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Students love us — and so will you.
4.9/5
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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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user