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Coordinate geometry

Learn How to Find the Equation of a Line and Midpoints Easily!

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Learn How to Find the Equation of a Line and Midpoints Easily!
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Emily Kelt

@emilykelt_yrng

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152 Followers

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This document covers key concepts in coordinate geometry, including collinearity, distance between points, midpoint formula, gradients, and equations of straight lines. It provides detailed explanations and examples for calculating distances, finding midpoints, determining gradients, and deriving line equations. The material is particularly useful for students learning about equations of straight lines, how to find midpoint coordinates, and gradient of perpendicular bisector calculations.

01/04/2023

104

Collinearity Are collin ear? Distance MAB Common Distance Midpoint d= X: the ÿ: m=tane what makes positive Live = Find A (1,7) тав = = = For

View

Perpendicular Lines and Equations of Straight Lines

This page delves into the properties of perpendicular lines and the methods for finding equations of straight lines.

Key concepts covered include:

  • Gradients of perpendicular lines
  • General equation of a straight line
  • Point-slope form of a line equation

Vocabulary: Perpendicular lines are lines that intersect at right angles (90 degrees).

Formula: The equation of a straight line formula is y = mx + c, where m is the gradient and c is the y-intercept.

Highlight: For perpendicular lines, the product of their gradients is always -1.

Example: Given points A(4,7) and B(3,-10), the gradient of line AB is calculated as m = (7-(-10))/(4-3) = 17.

The page also demonstrates how to find the equation of a line using the point-slope form: y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the gradient.

Example: For line CD with gradient -1/3 passing through point D(1,5), the equation is derived as: y - 5 = -1/3(x - 1), which simplifies to x + 3y - 16 = 0.

Collinearity Are collin ear? Distance MAB Common Distance Midpoint d= X: the ÿ: m=tane what makes positive Live = Find A (1,7) тав = = = For

View

Perpendicular Bisectors

This page focuses on perpendicular bisectors and their applications in triangle geometry.

The main topics covered are:

  • Finding the equation of a perpendicular bisector
  • Types of lines in triangles (altitudes, medians)
  • Handling vertical and horizontal lines

Definition: A perpendicular bisector is a line that passes through the midpoint of a line segment at a right angle.

Example: To find the equation of perpendicular bisector of line AB where A(5,-1) and B(13,23), first calculate the midpoint (9,11), then use the perpendicular gradient to form the equation: y - 11 = -3/8(x - 9).

Highlight: When dealing with vertical lines, the gradient is undefined, and the equation takes the form x = constant.

The page also introduces the concept of altitudes in triangles, which are perpendicular lines from a vertex to the opposite side.

Vocabulary: An altitude of a triangle is a line segment from a vertex perpendicular to the line containing the opposite side (or its extension).

Collinearity Are collin ear? Distance MAB Common Distance Midpoint d= X: the ÿ: m=tane what makes positive Live = Find A (1,7) тав = = = For

View

Medians and Triangle Centers

This page explores medians in triangles and various triangle centers.

Key concepts include:

  • Finding the equation of a median
  • Locating the centroid of a triangle
  • Relationships between different triangle centers

Definition: A median is a line segment that connects a vertex to the midpoint of the opposite side in a triangle.

Example: In triangle ABC with A(-5,6), B(-4,-10), and C(3,12), to find the equation of median BD, first calculate the midpoint of AC (-1,9), then use the point-slope form with B(-4,-10) to get the equation: 3y = 19x + 46.

Highlight: The centroid of a triangle is the point where all three medians intersect.

The page also mentions other important triangle centers:

  • Orthocenter: The point where all three altitudes intersect
  • Circumcenter: The point where all three perpendicular bisectors of the sides intersect
Collinearity Are collin ear? Distance MAB Common Distance Midpoint d= X: the ÿ: m=tane what makes positive Live = Find A (1,7) тав = = = For

View

Advanced Line Problems and Triangle Centers

This final page covers more complex problems involving lines and triangle centers.

Topics addressed include:

  • Finding equations of lines making specific angles with the x-axis
  • Solving for intersection points of perpendicular bisectors
  • Locating and understanding the significance of triangle centers

Example: To find the equation of a line AB making a 45° angle with the positive x-axis direction, use m = tan 45° = 1, resulting in the equation y = x - 3.

Highlight: The intersection point of perpendicular bisectors in a triangle is significant as it represents the circumcenter, which is equidistant from all three vertices.

The page concludes with a problem involving finding the intersection point of two lines, demonstrating the use of simultaneous equations to solve for the coordinates (8,5).

Vocabulary: The centroid of a triangle divides each median in the ratio 2:1, with the centroid being closer to the midpoint of the side.

This comprehensive guide provides a thorough understanding of how to find the equation of a straight line with one point, how to find midpoint coordinates on a graph, and various aspects of perpendicular bisector equations and calculations.

Collinearity Are collin ear? Distance MAB Common Distance Midpoint d= X: the ÿ: m=tane what makes positive Live = Find A (1,7) тав = = = For

View

Collinearity and Distance

This page introduces fundamental concepts in coordinate geometry, including collinearity and distance between points.

The page covers:

  • Determining if points are collinear
  • Calculating the distance between two points using the distance formula
  • Finding the midpoint of a line segment

Definition: Collinearity refers to points that lie on the same straight line.

Formula: The distance formula is used to calculate the length of a line segment: d = √[(x₂-x₁)² + (y₂-y₁)²]

Example: For points A(1,7) and B(5,-3), the distance AB is calculated as √[(5-1)² + (-3-7)²] = √[16 + 100] = √116 ≈ 10.77 units.

Highlight: The midpoint formula is crucial for finding the coordinates of the point exactly in the middle of a line segment: (x₁+x₂)/2, (y₁+y₂)/2

The page also touches on the concept of gradient and its relationship to the angle a line makes with the x-axis, introducing the formula m = tan θ.

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Know Coordinate geometry thumbnail

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Coordinate geometry

Notes on WJEC AS Pure Mathematics - Coordinate geometry.

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Subjects

/

Maths

/

Pure maths

/

Coordinate geometry

Learn How to Find the Equation of a Line and Midpoints Easily!

user profile picture

Emily Kelt

@emilykelt_yrng

·

152 Followers

Follow

This document covers key concepts in coordinate geometry, including collinearity, distance between points, midpoint formula, gradients, and equations of straight lines. It provides detailed explanations and examples for calculating distances, finding midpoints, determining gradients, and deriving line equations. The material is particularly useful for students learning about equations of straight lines, how to find midpoint coordinates, and gradient of perpendicular bisector calculations.

01/04/2023

104

 

S5

 

Maths

4

Collinearity Are collin ear? Distance MAB Common Distance Midpoint d= X: the ÿ: m=tane what makes positive Live = Find A (1,7) тав = = = For

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Access to all documents

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Join milions of students

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Perpendicular Lines and Equations of Straight Lines

This page delves into the properties of perpendicular lines and the methods for finding equations of straight lines.

Key concepts covered include:

  • Gradients of perpendicular lines
  • General equation of a straight line
  • Point-slope form of a line equation

Vocabulary: Perpendicular lines are lines that intersect at right angles (90 degrees).

Formula: The equation of a straight line formula is y = mx + c, where m is the gradient and c is the y-intercept.

Highlight: For perpendicular lines, the product of their gradients is always -1.

Example: Given points A(4,7) and B(3,-10), the gradient of line AB is calculated as m = (7-(-10))/(4-3) = 17.

The page also demonstrates how to find the equation of a line using the point-slope form: y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the gradient.

Example: For line CD with gradient -1/3 passing through point D(1,5), the equation is derived as: y - 5 = -1/3(x - 1), which simplifies to x + 3y - 16 = 0.

Collinearity Are collin ear? Distance MAB Common Distance Midpoint d= X: the ÿ: m=tane what makes positive Live = Find A (1,7) тав = = = For

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

Perpendicular Bisectors

This page focuses on perpendicular bisectors and their applications in triangle geometry.

The main topics covered are:

  • Finding the equation of a perpendicular bisector
  • Types of lines in triangles (altitudes, medians)
  • Handling vertical and horizontal lines

Definition: A perpendicular bisector is a line that passes through the midpoint of a line segment at a right angle.

Example: To find the equation of perpendicular bisector of line AB where A(5,-1) and B(13,23), first calculate the midpoint (9,11), then use the perpendicular gradient to form the equation: y - 11 = -3/8(x - 9).

Highlight: When dealing with vertical lines, the gradient is undefined, and the equation takes the form x = constant.

The page also introduces the concept of altitudes in triangles, which are perpendicular lines from a vertex to the opposite side.

Vocabulary: An altitude of a triangle is a line segment from a vertex perpendicular to the line containing the opposite side (or its extension).

Collinearity Are collin ear? Distance MAB Common Distance Midpoint d= X: the ÿ: m=tane what makes positive Live = Find A (1,7) тав = = = For

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

Medians and Triangle Centers

This page explores medians in triangles and various triangle centers.

Key concepts include:

  • Finding the equation of a median
  • Locating the centroid of a triangle
  • Relationships between different triangle centers

Definition: A median is a line segment that connects a vertex to the midpoint of the opposite side in a triangle.

Example: In triangle ABC with A(-5,6), B(-4,-10), and C(3,12), to find the equation of median BD, first calculate the midpoint of AC (-1,9), then use the point-slope form with B(-4,-10) to get the equation: 3y = 19x + 46.

Highlight: The centroid of a triangle is the point where all three medians intersect.

The page also mentions other important triangle centers:

  • Orthocenter: The point where all three altitudes intersect
  • Circumcenter: The point where all three perpendicular bisectors of the sides intersect
Collinearity Are collin ear? Distance MAB Common Distance Midpoint d= X: the ÿ: m=tane what makes positive Live = Find A (1,7) тав = = = For

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

Advanced Line Problems and Triangle Centers

This final page covers more complex problems involving lines and triangle centers.

Topics addressed include:

  • Finding equations of lines making specific angles with the x-axis
  • Solving for intersection points of perpendicular bisectors
  • Locating and understanding the significance of triangle centers

Example: To find the equation of a line AB making a 45° angle with the positive x-axis direction, use m = tan 45° = 1, resulting in the equation y = x - 3.

Highlight: The intersection point of perpendicular bisectors in a triangle is significant as it represents the circumcenter, which is equidistant from all three vertices.

The page concludes with a problem involving finding the intersection point of two lines, demonstrating the use of simultaneous equations to solve for the coordinates (8,5).

Vocabulary: The centroid of a triangle divides each median in the ratio 2:1, with the centroid being closer to the midpoint of the side.

This comprehensive guide provides a thorough understanding of how to find the equation of a straight line with one point, how to find midpoint coordinates on a graph, and various aspects of perpendicular bisector equations and calculations.

Collinearity Are collin ear? Distance MAB Common Distance Midpoint d= X: the ÿ: m=tane what makes positive Live = Find A (1,7) тав = = = For

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

Collinearity and Distance

This page introduces fundamental concepts in coordinate geometry, including collinearity and distance between points.

The page covers:

  • Determining if points are collinear
  • Calculating the distance between two points using the distance formula
  • Finding the midpoint of a line segment

Definition: Collinearity refers to points that lie on the same straight line.

Formula: The distance formula is used to calculate the length of a line segment: d = √[(x₂-x₁)² + (y₂-y₁)²]

Example: For points A(1,7) and B(5,-3), the distance AB is calculated as √[(5-1)² + (-3-7)²] = √[16 + 100] = √116 ≈ 10.77 units.

Highlight: The midpoint formula is crucial for finding the coordinates of the point exactly in the middle of a line segment: (x₁+x₂)/2, (y₁+y₂)/2

The page also touches on the concept of gradient and its relationship to the angle a line makes with the x-axis, introducing the formula m = tan θ.

Similar content

Maths - Coordinate geometry

Notes on WJEC AS Pure Mathematics - Coordinate geometry.

6

99

0

Know Coordinate geometry thumbnail

Can't find what you're looking for? Explore other subjects.

Knowunity is the #1 education app in five European countries

Knowunity has been named a featured story on Apple and has regularly topped the app store charts in the education category in Germany, Italy, Poland, Switzerland, and the United Kingdom. Join Knowunity today and help millions of students around the world.

Ranked #1 Education App

Download in

Google Play

Download in

App Store

Knowunity is the #1 education app in five European countries

4.9+

Average app rating

15 M

Pupils love Knowunity

#1

In education app charts in 12 countries

950 K+

Students have uploaded notes

Still not convinced? See what other students are saying...

iOS User

I love this app so much, I also use it daily. I recommend Knowunity to everyone!!! I went from a D to an A with it :D

Philip, iOS User

The app is very simple and well designed. So far I have always found everything I was looking for :D

Lena, iOS user

I love this app ❤️ I actually use it every time I study.