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Collinear Points

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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

Beginner

Start here! Easy to understand

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.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

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?

Real-World Problem

Question Exercise

Intermediate

Coordinate Check

Given the points P(0,1), Q(2,3), and R(4,5) on the coordinate plane, determine if they are collinear.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

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

Challenge Quiz

Single Choice Quiz

Advanced

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?

Recap

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Review key concepts and takeaways