Ever wondered how to prove three points lie on a...
Maths509 views·Updated 31 Jul 2026·4 pagesUnderstanding Straight Lines: Simple Notes and Examples
hana@hanajess777
1 / 4
1
of 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.
2
of 4
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 ) unless specifically asked to round.
Pro Tip: Remember that perpendicular lines meet at 90°, so their gradients are negative reciprocals of each other.
3
of 4
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.
4
of 4
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...
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.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Similar content
Most popular content in Maths
9MathsComprehensive Maths Concepts
Explore essential mathematical concepts including powers, geometry, statistics, and probability. This resource features 65 pages of detailed explanations, diagrams, and examples to enhance your understanding of topics such as right triangles, volume calculations, and data representation. Ideal for students seeking to strengthen their numeracy skills and grasp complex mathematical principles.
1180,1386,322
M
MathsMedium Level alerbra
Master challenging maths concepts with this medium level flashcard set designed for grade 7/8 students. Strengthen your problem-solving skills and boost your confidence in maths!
78913
MathsGCSE Maths (Higher) // Revision Guide
The only GCSE maths (higher) revision guide you need to get a grade 9! Contains every topic, each with all potential question types and their solutions.
102,62462
M
MathsMastering Maths: Essential Concepts for Grade 10
Boost your math skills with this comprehensive flashcard set covering key concepts for grade 10. Perfect for exam preparation and building a strong foundation in mathematics.
115641
M
MathsMastering Medium-Level Maths: Essential Flashcards for Grade 11 Students
Boost your Maths skills with this comprehensive set of flashcards designed specifically for Grade 11 students. Covering medium-level topics, these cards will help you ace your exams and build a solid foundation for advanced Maths.
119743
M
MathsMaths Made Easy: Essential Concepts for Grade 7
Master key mathematical concepts with this comprehensive flashcard set designed specifically for 13-year-old students. Strengthen your understanding and ace your exams!
77922
P
MathsPercentage,fractions and decimals
how well do you know percentages,fractions and decimals
73213
M
Mathsmaths SOHCAHTOA
Trigonometric ratios SOHCAHTOA for calculating angles and sides in right-angled triangles.
112240
MathsGCSE Maths Revision Guide
Comprehensive solutions and explanations for past GCSE Maths Paper 1 questions. Topics include solving quadratics, area calculations, ratios, probability, and more. Ideal for students preparing for their exams, with clear step-by-step methods and key concepts highlighted.
111,42317
Most popular content
9
SociologySociology of Families: Comprehensive Revision
Dive into an extensive overview of family dynamics, perspectives, and patterns in sociology. This resource covers key concepts such as family diversity, gender roles, marriage, and the impact of social policies on family structures. Perfect for A-Level Sociology students preparing for Paper 2.
1273,7982,306
SociologySociology of Education Overview
Explore comprehensive A-Level Sociology notes on the education system, covering key theories, policies, and sociological perspectives. This resource includes insights on marketisation, gender roles, cultural deprivation, and educational inequalities, providing a thorough understanding of how education shapes social stratification and individual achievement. Ideal for exam preparation and in-depth study.
12102,9863,041
CriminologyCriminology: Crime & Punishment Overview
Comprehensive mindmaps covering key concepts in the Crime and Punishment topic for WJEC Criminology Unit 4. This resource includes detailed insights into the Criminal Justice System, crime prevention strategies, sentencing models, and the roles of various agencies. Ideal for A-Level revision, ensuring you grasp essential theories and legislative processes to excel in your exams.
1254,8981,061
SociologyComprehensive Crime & Deviance Overview
Explore an extensive revision of crime and deviance topics, including theories, types of crime, and the impact of media. This resource covers key concepts such as Marxism, functionalism, gender and crime, and the influence of globalization on criminal behavior. Ideal for students seeking a thorough understanding of criminology and its various theories. Type: Full Topic Revision.
1251,6721,401
C
BiologyCell Biology and Cell structure
cell structures
113,2990
CriminologyWJEC Unit 4 Criminology
Criminology unit 4 detailed revision note
127,177125
SociologySociological Theories Overview
Comprehensive revision of key sociological theories including Functionalism, Marxism, Feminism, and Interpretivism. Explore concepts like value freedom, identity formation, and the critique of social control. Ideal for AQA A-Level Sociology students preparing for exams. This summary covers essential theories and their implications in sociology, providing a clear understanding of each perspective.
1231,494846
T
BiologyThe functions of subcellular structures - B1 Biology
Flashcards on the different functions of subcellular structures: cell membrane, nucleus, mitochondria, ribosomes, cytoplasm, permant vacuole, chloroplasts and cell wall.
112,8215
1
Biology1.cells Gcse biology question cards
combined science higher biology
112,3734
Students love us — and so will you.
4.6/5App Store
4.7/5Google 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 SiOS 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 KlichAndroid 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.
AnnaiOS user
Maths509 views·Updated 31 Jul 2026·4 pagesUnderstanding 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...
1
of 4
Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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.
2
of 4Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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 ) unless specifically asked to round.
Pro Tip: Remember that perpendicular lines meet at 90°, so their gradients are negative reciprocals of each other.
3
of 4Sign up to see the content. It'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.
4
of 4Sign up to see the content. It'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...
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.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Similar content
Most popular content in Maths
9MathsComprehensive Maths Concepts
Explore essential mathematical concepts including powers, geometry, statistics, and probability. This resource features 65 pages of detailed explanations, diagrams, and examples to enhance your understanding of topics such as right triangles, volume calculations, and data representation. Ideal for students seeking to strengthen their numeracy skills and grasp complex mathematical principles.
1180,1386,322
M
MathsMedium Level alerbra
Master challenging maths concepts with this medium level flashcard set designed for grade 7/8 students. Strengthen your problem-solving skills and boost your confidence in maths!
78913
MathsGCSE Maths (Higher) // Revision Guide
The only GCSE maths (higher) revision guide you need to get a grade 9! Contains every topic, each with all potential question types and their solutions.
102,62462
M
MathsMastering Maths: Essential Concepts for Grade 10
Boost your math skills with this comprehensive flashcard set covering key concepts for grade 10. Perfect for exam preparation and building a strong foundation in mathematics.
115641
M
MathsMastering Medium-Level Maths: Essential Flashcards for Grade 11 Students
Boost your Maths skills with this comprehensive set of flashcards designed specifically for Grade 11 students. Covering medium-level topics, these cards will help you ace your exams and build a solid foundation for advanced Maths.
119743
M
MathsMaths Made Easy: Essential Concepts for Grade 7
Master key mathematical concepts with this comprehensive flashcard set designed specifically for 13-year-old students. Strengthen your understanding and ace your exams!
77922
P
MathsPercentage,fractions and decimals
how well do you know percentages,fractions and decimals
73213
M
Mathsmaths SOHCAHTOA
Trigonometric ratios SOHCAHTOA for calculating angles and sides in right-angled triangles.
112240
MathsGCSE Maths Revision Guide
Comprehensive solutions and explanations for past GCSE Maths Paper 1 questions. Topics include solving quadratics, area calculations, ratios, probability, and more. Ideal for students preparing for their exams, with clear step-by-step methods and key concepts highlighted.
111,42317
Most popular content
9SociologySociology of Families: Comprehensive Revision
Dive into an extensive overview of family dynamics, perspectives, and patterns in sociology. This resource covers key concepts such as family diversity, gender roles, marriage, and the impact of social policies on family structures. Perfect for A-Level Sociology students preparing for Paper 2.
1273,7982,306
SociologySociology of Education Overview
Explore comprehensive A-Level Sociology notes on the education system, covering key theories, policies, and sociological perspectives. This resource includes insights on marketisation, gender roles, cultural deprivation, and educational inequalities, providing a thorough understanding of how education shapes social stratification and individual achievement. Ideal for exam preparation and in-depth study.
12102,9863,041
CriminologyCriminology: Crime & Punishment Overview
Comprehensive mindmaps covering key concepts in the Crime and Punishment topic for WJEC Criminology Unit 4. This resource includes detailed insights into the Criminal Justice System, crime prevention strategies, sentencing models, and the roles of various agencies. Ideal for A-Level revision, ensuring you grasp essential theories and legislative processes to excel in your exams.
1254,8981,061
SociologyComprehensive Crime & Deviance Overview
Explore an extensive revision of crime and deviance topics, including theories, types of crime, and the impact of media. This resource covers key concepts such as Marxism, functionalism, gender and crime, and the influence of globalization on criminal behavior. Ideal for students seeking a thorough understanding of criminology and its various theories. Type: Full Topic Revision.
1251,6721,401
C
BiologyCell Biology and Cell structure
cell structures
113,2990
CriminologyWJEC Unit 4 Criminology
Criminology unit 4 detailed revision note
127,177125
SociologySociological Theories Overview
Comprehensive revision of key sociological theories including Functionalism, Marxism, Feminism, and Interpretivism. Explore concepts like value freedom, identity formation, and the critique of social control. Ideal for AQA A-Level Sociology students preparing for exams. This summary covers essential theories and their implications in sociology, providing a clear understanding of each perspective.
1231,494846
T
BiologyThe functions of subcellular structures - B1 Biology
Flashcards on the different functions of subcellular structures: cell membrane, nucleus, mitochondria, ribosomes, cytoplasm, permant vacuole, chloroplasts and cell wall.
112,8215
1
Biology1.cells Gcse biology question cards
combined science higher biology
112,3734
Students love us — and so will you.
4.6/5App Store
4.7/5Google 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 SiOS 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 KlichAndroid 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.
AnnaiOS user