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Coordinate Geometry

What is coordinate geometry?

  • Coordinate geometry is the study of geometric figures like lines and shapes, using coordinates.
  • Given two points, at GCSE, you are expected to know how to find;
    1. Midpoint of a Line
    2. Gradient of a Line
    3. Length of a Line

Midpoint of a Line

How do I find the midpoint of a line in two dimensions (2D)?

  • The midpoint of a line will be the same distance from both endpoints
  • You can think of a midpoint as being the average (mean) of two coordinates
  • The midpoint of open parentheses x subscript 1 comma space y subscript 1 close parentheses and open parentheses x subscript 2 comma space y subscript 2 close parentheses is

open parentheses fraction numerator x subscript 1 plus x subscript 2 over denominator 2 end fraction space comma space fraction numerator y subscript 1 plus y subscript 2 over denominator 2 end fraction close parentheses

How can I extend the idea of the midpoint in two dimensions (2D)?

  • The midpoint of AB splits AB in the ratio 1 : 1.
  • If you are asked to find the point that divides AB in the ratio m : n, then you need to find the point that lies fraction numerator m over denominator m plus n end fraction of the way from A to B.
    • E.g. dividing AB in the ratio 2 : 3 means finding the point that is 2 over 5 of the way from to B.
  • Normally an exam question will ask you to find the point that divides AB in the ratio 1 : n, so;
    • find the difference between the x coordinates,
    • divide this difference by [1 + n], and add the result to the x coordinate of A,
    • repeat for the y coordinates.

How do I find the midpoint of a line in three dimensions (3D)?

  • Finding the midpoint of a line in 3D involves a simple extension of the formula used for 2D midpoints
  • Similar to before, you can think of a midpoint as being the average (mean) of three x and three y coordinates
  • The midpoint of open parentheses x subscript 1 comma space y subscript 1 comma space z subscript 1 close parentheses and open parentheses x subscript 2 comma space y subscript 2 comma space z subscript 2 close parentheses is

open parentheses fraction numerator x subscript 1 plus x subscript 2 over denominator 2 end fraction space comma space fraction numerator y subscript 1 plus y subscript 2 over denominator 2 end fraction comma space fraction numerator z subscript 1 plus z subscript 2 over denominator 2 end fraction close parentheses

Exam Tip

  • If working in 2D (most questions!) making a quick sketch of the two points will help you know roughly where the midpoint should be, which can be helpful to check your answer
  • If working in 3D (some questions!), just check that your midpoint's x coordinate lies between the two given x coordinates, and so on for the y and z coordinates

Worked example

The coordinates of A are (−4, 3) and the coordinates of B are (8, −12).

(a)
Find M, the midpoint of AB.

The midpoint can be found using Mopen parentheses fraction numerator x subscript 1 plus x subscript 2 over denominator 2 end fraction space comma space fraction numerator y subscript 1 plus y subscript 2 over denominator 2 end fraction close parentheses
Fill in the values of x and y  from each coordinate

open parentheses fraction numerator negative 4 plus 8 over denominator 2 end fraction space comma space fraction numerator 3 plus negative 12 over denominator 2 end fraction close parentheses equals open parentheses 4 over 2 comma space fraction numerator negative 9 over denominator 2 end fraction close parentheses
Simplify
M = (2, −4.5)
(b)
A point N divides AB in the ratio 1 : 2.
Find the coordinates of N.

Calculate the difference between the x coordinate of and the x coordinate of
B

table row cell 8 minus open parentheses negative 4 close parentheses end cell equals 12 end table

Divide this difference by 3 (as there are 1 + 2 = 3 parts in the ratio 1 : 2)

table row cell 12 divided by 3 end cell equals 4 end table
Add 4 to the x coordinate of A
table row cell x space coordinate space of space N end cell equals cell negative 4 plus 4 equals 0 end cell end table
Repeat for y
table row cell negative 12 minus 3 end cell equals cell negative 15 end cell row cell open parentheses negative 15 close parentheses divided by 3 end cell equals cell negative 5 end cell row cell y space coordinate space of space N end cell equals cell 3 plus open parentheses negative 5 close parentheses equals negative 2 end cell end table

Alternatively, you may find it helpful to sketch the coordinates A and B and find N intuitively. Note that in the sketch, the coordinates do not need to be placed in the correct orientation to one another, simply along a straight line:

2-12-2-midpoints-we

Write the final answer as a coordinate point
N = (0, −2)

Gradient of a Line

What is the gradient of a line?

  • The gradient is a measure of how steep a 2D line is
    • A large value for the gradient means the line is steeper than for a small value of the gradient
      • A gradient of 3 is steeper than a gradient of 2
      • A gradient of −5 is steeper than a gradient of −4
    • A positive gradient means the line goes upwards from left to right
    • A negative gradient means the line goes downwards from left to right
  • In the equation for a straight line, y equals m x plus c, the gradient is represented by m
    • The gradient of y equals negative 3 x plus 2 is −3

How do I find the gradient of a line?

  • The gradient can be calculated using

gradient space equals space fraction numerator change space in space y over denominator change space in space x end fraction

  • You may see this written as rise over run instead
  • For two coordinates open parentheses x subscript 1 space comma space y subscript 1 close parentheses and open parentheses x subscript 2 space comma space y subscript 2 close parentheses the gradient of the line joining them is

fraction numerator y subscript 2 minus y subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction space or space fraction numerator y subscript 1 minus y subscript 2 over denominator x subscript 1 minus x subscript 2 end fraction

    • The order of the coordinates must be consistent on the top and bottom
    • i.e. (Point 1 – Point 2) or (Point 2 – Point 1) for both the top and bottom

How do I draw a line with a given gradient?

  • A line with a gradient of 4 could instead be written as 4 over 1. 
    • As gradient space equals space fraction numerator change space in space y over denominator change space in space x end fraction , this would mean for every 1 unit to the right (x direction), the line moves upwards (y direction) by 4 units.
    • Notice that 4 also equals fraction numerator negative 4 over denominator negative 1 end fraction, so for every 1 unit to the left, the line moves downwards by 4 units
  • If the gradient was −4, then rise over run equals fraction numerator negative 4 over denominator plus 1 end fraction or fraction numerator plus 4 over denominator negative 1 end fraction. This means the line would move downwards by 4 units for every 1 unit to the right.
  • If the gradient is a fraction, for example 2 over 3, we can think of this as either
    • For every 1 unit to the right, the line moves upwards by 2 over 3, or
    • For every 3 units to the right, the line moves upwards by 2.
    • (Or for every 3 units to the left, the line moves downwards by 2.)
  • If the gradient was negative 2 over 3 this would mean the line would move downwards by 2 units for every 3 units to the right
  • Once you know this, you can select a point (usually given, for example the y-intercept) and then count across and upwards or downwards to find another point on the line, and then join them with a straight line

Exam Tip

  • Be very careful with negative numbers when calculating the gradient; write down your working rather than trying to do it in your head to avoid mistakes
    • For example, fraction numerator open parentheses negative 3 close parentheses minus open parentheses negative 9 close parentheses over denominator open parentheses negative 18 close parentheses minus open parentheses 7 close parentheses end fraction

Worked example

(a)

Find the gradient of the line joining (-1, 4) and (7, 28)

Using gradient space equals space fraction numerator change space in space y over denominator change space in space x end fraction:

fraction numerator 28 minus 4 over denominator 7 minus negative 1 end fraction

Simplify: 

fraction numerator 28 minus 4 over denominator 7 minus negative 1 end fraction equals 24 over 8 equals 3

Gradient = 3

 

(b)

On the grid below, draw a line with gradient −2 that passes through (0, 1).

Mark the point (0, 1) and then count 2 units down for every 1 unit across

cie-igcse-gradients-of-lines-we-1

 

(c)begin mathsize 11px style table row blank row blank row blank end table end style

On the grid below, draw a line with gradient 2 over 3 that passes through (0,-1)

Mark the point (0,-1) and then count 2 units up for every 3 units across

cie-igcse-gradients-of-lines-we-2

Length of a Line

How do I calculate the length of a line?

  • The distance between two points with coordinates open parentheses x subscript 1 space comma space y subscript 1 close parentheses and open parentheses x subscript 2 space comma space y subscript 2 close parentheses can be found using the formula

d equals square root of open parentheses x subscript 1 minus x subscript 2 close parentheses squared plus open parentheses y subscript 1 minus y subscript 2 close parentheses squared end root

  • This formula is really just Pythagoras’ Theorem  a squared equals b squared plus c squared, applied to the difference in the x-coordinates and the difference in the y-coordinates;

Basic Coordinate Geometry Notes Diagram 2

  • You may be asked to find the length of a diagonal in 3D space. This can be answered using 3D Pythagoras

Exam Tip

  • As we are squaring the difference in and  in the formula, it does not matter if they are positive or negative
    • 32 is the same as (-3)2, this may help to speed up your working

Worked example

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

Calculate the distance of the line segment AB.

Using the formula for the distance between two points, d equals square root of open parentheses x subscript 1 minus x subscript 2 close parentheses squared plus open parentheses y subscript 1 minus y subscript 2 close parentheses squared end root 

Substituting in the two given coordinates:

d equals square root of open parentheses 3 minus negative 5 close parentheses squared plus open parentheses negative 4 minus 2 close parentheses squared end root

Simplify: 

d equals square root of open parentheses 8 close parentheses squared plus open parentheses negative 6 close parentheses squared end root space equals space square root of 64 plus 36 end root equals square root of 100 equals 10

Answer = 10 units

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