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Collinear Points

Study Guide

Key Definition

Three or more points that lie on the same line are called collinear points.

Important Notes

  • If points A, B, and C are collinear, then point B is between A and C if and only if AB + BC = AC.

Mathematical Notation

Use capital letters for points (e.g., A, B, C).Denote segments as \u0305AB or vectors as \u2192AB.Denote the line through A and B as AB↔.Remember to use proper notation when solving problems

Why It Works

Algebraically, three points A(x₁,y₁), B(x₂,y₂), and C(x₃,y₃) are collinear if the area of the triangle they form is zero: x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)=0. Equivalently, the slopes between each pair of points are equal.

Remember

Not all points are collinear. If no single line can be drawn through all points, the points are not collinear.

Quick Reference

Collinear Points:Three or more points that lie on the same line

Understanding Collinear Points

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Video explanation of this concept

concept. Use space or enter to play video.concept thumbnailBeginner

Start here! Easy to understand

BeginnerIntermediateAdvanced

Beginner Explanation

Given two points A(x₁,y₁) and B(x₂,y₂), the slope m of line AB is (y₂−y₁)/(x₂−x₁). Three points A, B, and C are collinear if slope AB equals slope BC.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

Given the points A(1,2), B(2,4), C(3,6); D(0,0), E(1,2), F(2,5); G(0,1), H(1,3), I(2,6); J(1,0), K(2,1), L(3,1), which of the following triples are collinear?

AA, B, CBD, E, FCG, H, IDJ, K, LCheck AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Coordinate Check

Given the points P(0,1), Q(2,3), and R(4,5) on the coordinate plane, determine if they are collinear.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

Can you think of a situation where it would be important to determine if points are collinear?

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

Given the points A(0,0), B(1,1), C(2,2); D(2,2), E(3,3), F(4,4); G(−1,−1), H(0,0), I(1,1); J(0,1), K(1,2), L(2,4), which of the following sets of points are not collinear?

AA, B, CBD, E, FCG, H, IDJ, K, LCheck AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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