Coordinate Geometry | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Introduction

• Coordinate geometry involves studying geometric figures such as lines and shapes by using coordinates.
• At the GCSE level, you are required to understand how to determine various aspects when given two points:
  • Midpoint of a Line: The midpoint of a line segment is the point that divides the line segment into two equal parts.
  • Gradient of a Line: The gradient of a line indicates how steep the line is. It's the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
  • Length of a Line: This refers to the distance between two points on the line, typically calculated using the Pythagorean theorem.

Midpoint of a Line

• The midpoint of a line is equidistant from both endpoints. 
• To visualize, consider it as the average (mean) of the two coordinates.
• The midpoint of (x₁, y₁) and (x₂, y₂) is

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

• The midpoint of line segment AB divides it in the ratio 1 : 1.
• When asked to find a point that divides AB in the ratio m : n, locate the point that lies  of the way from A to B. 
  • For example, dividing AB in the ratio 2 : 3 means finding a point that is 2/5 of the way from A 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  (x₁, y₁, z₁) and (x₂, y₂, z₂) is
  

Gradient of a Line

• The gradient of a line indicates how steep it is in a 2D space.
  • A larger gradient value signifies a steeper line compared to a smaller gradient value.
    • For example, 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 implies the line rises from left to right.
  • A negative gradient implies the line descends from left to right.
• In the equation for a straight line, y = mx + c, the gradient is represented by m
  • The gradient of y equals - 3x + 2 is −3

How do I find the gradient of a line?

• The gradient can be calculated using the formula:
  
• You may see this written as instead
• For two coordinates (x₁, y₁) and (x₂, y₂), the gradient of the line joining them is:
  
• 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 4/1.
  •  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 -4/-1, so for every 1 unit to the left, the line moves downwards by 4 units
• If the gradient was −4, then 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 2/3 we can think of this as either
  • For every 1 unit to the right, the line moves upwards by 2/3 , 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 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

Length of a Line

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

• This formula is really just Pythagoras’ Theorem  a² = b² + c² squared, applied to the difference in the x-coordinates and the difference in the y-coordinates;

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