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

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Emily Kelt

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Coordinate geometry

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

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

117

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 90degrees90 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 A4,74,7 and B3,−103,-10, the gradient of line AB is calculated as m = 7−(−107-(-10)/4−34-3 = 17.

The page also demonstrates how to find the equation of a line using the point-slope form: y - y₁ = mx−x1x - x₁, where x1,y1x₁, y₁ is a point on the line and m is the gradient.

Example: For line CD with gradient -1/3 passing through point D1,51,5, the equation is derived as: y - 5 = -1/3x−1x - 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,mediansaltitudes, 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 A5,−15,-1 and B13,2313,23, first calculate the midpoint 9,119,11, then use the perpendicular gradient to form the equation: y - 11 = -3/8x−9x - 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 oritsextensionor 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−5,6-5,6, B−4,−10-4,-10, and C3,123,12, to find the equation of median BD, first calculate the midpoint of AC −1,9-1,9, then use the point-slope form with B−4,−10-4,-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 8,58,5.

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.

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

user profile picture

Emily Kelt

@emilykelt_yrng

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... Show more

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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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 90degrees90 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 A4,74,7 and B3,−103,-10, the gradient of line AB is calculated as m = 7−(−107-(-10)/4−34-3 = 17.

The page also demonstrates how to find the equation of a line using the point-slope form: y - y₁ = mx−x1x - x₁, where x1,y1x₁, y₁ is a point on the line and m is the gradient.

Example: For line CD with gradient -1/3 passing through point D1,51,5, the equation is derived as: y - 5 = -1/3x−1x - 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

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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,mediansaltitudes, 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 A5,−15,-1 and B13,2313,23, first calculate the midpoint 9,119,11, then use the perpendicular gradient to form the equation: y - 11 = -3/8x−9x - 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 oritsextensionor 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 contentIt's free!

Access to all documents

Improve your grades

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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−5,6-5,6, B−4,−10-4,-10, and C3,123,12, to find the equation of median BD, first calculate the midpoint of AC −1,9-1,9, then use the point-slope form with B−4,−10-4,-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

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Access to all documents

Improve your grades

Join milions of students

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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 8,58,5.

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

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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(x₂-x₁)² + (y₂-y₁)²

Example: For points A1,71,7 and B5,−35,-3, the distance AB is calculated as √(5−1)2+(−3−7)2(5-1)² + (-3-7)² = √16+10016 + 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+x2x₁+x₂/2, y1+y2y₁+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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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

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

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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

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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

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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

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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

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