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IB DP Maths: AI SL

Revision Notes

Home / DP / Maths: AI SL / IB / Revision Notes / 3. Geometry & Trigonometry / 3.1 Geometry Toolkit / 3.1.1 Coordinate Geometry


3.1.1 Coordinate Geometry


Basic Coordinate Geometry

What are cartesian coordinates?

  • Cartesian coordinates are basically the x-y coordinate system
    • They allow us to label where things are in a two-dimensional plane
  • In the 2D cartesian system, the horizontal axis is labelled x and the vertical axis is labelled y

 

What can we do with coordinates?

  • If we have two points with coordinates (x1 , y1) and (x2 , y2) then we should be able to find
    • The midpoint of the two points
    • The distance between the two points
    • The gradient of the line between them

 

How do I find the midpoint of two points?

  • The midpoint is the average (middle) point
    • It can be found by finding the middle of the x-coordinates and the middle of the y-coordinates
  • The coordinates of the midpoint will be

begin mathsize 22px style open parentheses fraction numerator x subscript 1 plus blank x subscript 2 over denominator 2 end fraction blank comma blank fraction numerator y subscript 1 plus blank y subscript 2 over denominator 2 end fraction blank close parentheses end style

    • This is given in the formula booklet under the prior learning section at the beginning

Basic Coordinate Geometry Notes Diagram 5

How do I find the distance between two points?

  • The distance between two points with coordinates (x1 , y1) and (x2 , y2) can be found using the formula

begin mathsize 22px style d space equals blank square root of open parentheses x subscript 1 minus blank x subscript 2 close parentheses squared space plus space open parentheses y subscript 1 minus blank y subscript 2 close parentheses squared end root blank end style

    • This is given in the formula booklet in the prior learning section at the beginning
  • Pythagoras’ Theorem a squared space equals space b squared plus c squared  is used to find the length of a line between two coordinates
  • If the coordinates are labelled A and B then the line segment between them is written with the notation [AB]

Basic Coordinate Geometry Notes Diagram 2

How do I find the gradient of the line between two points?

  • The gradient of a line between two points with coordinates (x1 , y1) and (x2 , y2) can be found using the formula

begin mathsize 22px style m equals blank fraction numerator y subscript 2 minus blank y subscript 1 over denominator x subscript 2 minus blank x subscript 1 end fraction end style

    • This is given in the formula booklet under section 2.1 Gradient formula
    • This is usually known as  m equals rise over run

Worked Example

Point A has coordinates (3, 4) and point B has coordinates (-5, 2).

i)
Calculate the distance of the line segment AB.

ai-sl-3-1-1-basic-cg-we-i

ii)
Find the gradient of the line connecting points A and B.

 ai-sl-3-1-1-basic-cg-we-ii

iii)
Find the midpoint of [AB ] .

ai-sl-3-1-1-basic-cg-we-iii

Perpendicular Bisectors

What is a perpendicular bisector?

  • A perpendicular bisector of a line segment cuts the line segment in half at a right angle
    • Perpendicular lines meet at right angles
    • Bisector means to cut in half
  • Two lines are perpendicular if the product of their gradients is -1

 

How do I find the equation of the perpendicular bisector of a line segment?

  • To find the equation of a straight line you need to find
    • The gradient of the line
    • A coordinate of a point on the line
  • To find the equation of the perpendicular bisector of a line segment follow the steps:
    • STEP 1: Find the coordinates of the midpoint of the line segment
      • We know that the perpendicular bisector will cut the line segment in half so we can use the midpoint of the line segment as the known coordinate on the bisector
    • STEP 2: Find the gradient of the line segment
    • STEP 3: Find the gradient of the perpendicular bisector
      • This will be -1 divided by the gradient of the line segment
    • STEP 4: Substitute the gradient of the perpendicular bisector and the coordinates of the midpoint into an equation for a straight line
      • The point-gradient form bold italic y minus blank bold italic y subscript 1 blank end subscript equals bold italic m bold space left parenthesis bold italic x minus blank bold italic x subscript 1 right parenthesisis the easiest
    • STEP 5: Rearrange into the required form
      • Either y space equals space m x space plus space c  or  a x space plus space b y space plus thin space d space equals space 0 
      • These equations for a straight line are given in the formula booklet

Worked Example

Point A has coordinates (4, -6) and point B has coordinates (8, 6).  Find the equation of the perpendicular bisector to [AB ]. Give your answer in the form a x space plus space b y space plus space d space equals space 0.

ai-sl-3-1-1-perp-bis-we



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Exponents & Logarithms
          • 1.1.3 Approximation & Estimation
            • 1.1.4 GDC: Solving Equations
            • 1.2 Sequences & Series
              • 1.2.1 Language of Sequences & Series
                • 1.2.2 Arithmetic Sequences & Series
                  • 1.2.3 Geometric Sequences & Series
                    • 1.2.4 Applications of Sequences & Series
                    • 1.3 Financial Applications
                      • 1.3.1 Compound Interest & Depreciation
                        • 1.3.2 Amortisation & Annuities
                      • 2. Functions
                        • 2.1 Linear Functions & Graphs
                          • 2.1.1 Equations of a Straight Line
                          • 2.2 Further Functions & Graphs
                            • 2.2.1 Functions
                              • 2.2.2 Graphing Functions
                                • 2.2.3 Properties of Graphs
                                • 2.3 Modelling with Functions
                                  • 2.3.1 Linear & Piecewise Models
                                    • 2.3.2 Quadratic & Cubic Models
                                      • 2.3.3 Exponential Models
                                        • 2.3.4 Direct & Inverse Variation
                                          • 2.3.5 Sinusoidal Models
                                            • 2.3.6 Strategy for Modelling Functions
                                          • 3. Geometry & Trigonometry
                                            • 3.1 Geometry Toolkit
                                              • 3.1.1 Coordinate Geometry
                                                • 3.1.2 Arcs & Sectors
                                                • 3.2 Geometry of 3D Shapes
                                                  • 3.2.1 3D Coordinate Geometry
                                                    • 3.2.2 Volume & Surface Area
                                                    • 3.3 Trigonometry
                                                      • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                        • 3.3.2 Non Right-Angled Trigonometry
                                                          • 3.3.3 Applications of Trigonometry & Pythagoras
                                                          • 3.4 Voronoi Diagrams
                                                            • 3.4.1 Voronoi Diagrams
                                                              • 3.4.2 Toxic Waste Dump Problem
                                                            • 4. Statistics & Probability
                                                              • 4.1 Statistics Toolkit
                                                                • 4.1.1 Sampling & Data Collection
                                                                  • 4.1.2 Statistical Measures
                                                                    • 4.1.3 Frequency Tables
                                                                      • 4.1.4 Linear Transformations of Data
                                                                        • 4.1.5 Outliers
                                                                          • 4.1.6 Univariate Data
                                                                            • 4.1.7 Interpreting Data
                                                                            • 4.2 Correlation & Regression
                                                                              • 4.2.1 Bivariate data
                                                                                • 4.2.2 Correlation Coefficients
                                                                                  • 4.2.3 Linear Regression
                                                                                  • 4.3 Probability
                                                                                    • 4.3.1 Probability & Types of Events
                                                                                      • 4.3.2 Conditional Probability
                                                                                        • 4.3.3 Sample Space Diagrams
                                                                                        • 4.4 Probability Distributions
                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                            • 4.4.2 Expected Values
                                                                                            • 4.5 Binomial Distribution
                                                                                              • 4.5.1 The Binomial Distribution
                                                                                                • 4.5.2 Calculating Binomial Probabilities
                                                                                                • 4.6 Normal Distribution
                                                                                                  • 4.6.1 The Normal Distribution
                                                                                                    • 4.6.2 Calculations with Normal Distribution
                                                                                                    • 4.7 Hypothesis Testing
                                                                                                      • 4.7.1 Hypothesis Testing
                                                                                                        • 4.7.2 Chi-squared Test for Independence
                                                                                                          • 4.7.3 Goodness of Fit Test
                                                                                                            • 4.7.4 The t-test
                                                                                                          • 5. Calculus
                                                                                                            • 5.1 Differentiation
                                                                                                              • 5.1.1 Introduction to Differentiation
                                                                                                                • 5.1.2 Applications of Differentiation
                                                                                                                  • 5.1.3 Modelling with Differentiation
                                                                                                                  • 5.2 Integration
                                                                                                                    • 5.2.1 Trapezoid Rule: Numerical Integration
                                                                                                                      • 5.2.2 Introduction to Integration
                                                                                                                        • 5.2.3 Applications of Integration


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                                                                                                                      Author: Amber

                                                                                                                      Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.


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