Mathematics (4024)
Topic 9 of 18Cambridge O Levels

Coordinate Geometry

Analyzing lines and shapes on a Cartesian plane using algebraic formulas.

Coordinate Geometry is a branch of mathematics that serves as a bridge between algebra and geometry. It allows us to study geometric shapes by placing them on a Cartesian plane (also known as the coordinate plane). This plane is defined by two perpendicular number lines: the horizontal x-axis and the vertical y-axis. The point where they intersect is called the origin (0, 0). Any point on this plane can be uniquely identified by an ordered pair of numbers called coordinates (x, y).


### 1. Calculating the Length of a Line Segment

To find the distance between two points, A(x₁, y₁) and B(x₂, y₂), we can imagine a right-angled triangle with the line segment AB as its hypotenuse. The horizontal side of the triangle has a length of (x₂ - x₁), and the vertical side has a length of (y₂ - y₁). Using Pythagoras' theorem (a² + b² = c²), we derive the distance formula.


The formula for the length (or distance) of a line segment is:

Length = √[(x₂ - x₁)² + (y₂ - y₁)²]


Process:

  • Identify the coordinates of the two points, (x₁, y₁) and (x₂, y₂).
  • Subtract the x-coordinates and square the result.
  • Subtract the y-coordinates and square the result.
  • Add these two squared values together.
  • Take the square root of the sum to find the length.

    • *Example:* Find the distance between P(2, 3) and Q(8, 11).

    Length PQ = √[(8 - 2)² + (11 - 3)²] = √[6² + 8²] = √[36 + 64] = √100 = 10 units.


    ### 2. Finding the Midpoint of a Line Segment

    The midpoint of a line segment is the point that is exactly halfway between the two endpoints. Its coordinates are simply the average of the x-coordinates and the average of the y-coordinates of the endpoints.


    The formula for the midpoint M of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is:

    M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )


    Process:

  • Add the x-coordinates of the two points and divide by 2.
  • Add the y-coordinates of the two points and divide by 2.
  • Write the results as a new coordinate pair (x, y).

    • *Example:* Find the midpoint of the line connecting A(1, 9) and B(7, 3).

    Midpoint M = ( (1 + 7)/2 , (9 + 3)/2 ) = ( 8/2 , 12/2 ) = (4, 6).


    ### 3. Calculating the Gradient of a Line Segment

    The gradient (often denoted by 'm') of a line measures its steepness or slope. It is calculated as the ratio of the vertical change ("rise") to the horizontal change ("run") between any two points on the line.


    The formula for the gradient (m) is:

    m = (y₂ - y₁)/(x₂ - x₁) = change in y / change in x


  • A positive gradient means the line slopes upwards from left to right.
  • A negative gradient means the line slopes downwards from left to right.
  • A gradient of zero indicates a horizontal line (y₁ = y₂).
  • An undefined gradient indicates a vertical line (x₁ = x₂).

    • Furthermore, gradients help define the relationship between lines:

  • Parallel lines have the same gradient (m₁ = m₂).
  • Perpendicular lines have gradients whose product is -1 (m₁ × m₂ = -1). This means one gradient is the negative reciprocal of the other (e.g., if m₁ = 2, then m₂ = -1/2).

    • ### 4. Finding the Equation of a Straight Line

    The equation of a straight line can be expressed in the gradient-intercept form:

    y = mx + c

    Here, m is the gradient and c is the y-intercept (the y-coordinate where the line crosses the y-axis).


    Process to find the equation:

  • Find the gradient (m) using two points if not already given.
  • Substitute 'm' into the equation: y = (value of m)x + c.
  • Find the y-intercept (c) by substituting the coordinates of one of the known points (x, y) into the equation and solving for 'c'.
  • Write the final equation with the now-known values of 'm' and 'c'.

    • *Example:* Find the equation of the line passing through (2, 5) and (4, 11).

    Step 1: Find the gradient.

    m = (11 - 5) / (4 - 2) = 6 / 2 = 3.

    Step 2: Substitute m into the equation.

    y = 3x + c.

    Step 3: Use a point, say (2, 5), to find c.

    5 = 3(2) + c

    5 = 6 + c

    c = -1.

    Step 4: Write the final equation.

    y = 3x - 1

    Key Points to Remember

    • 1The Cartesian plane uses a horizontal x-axis and vertical y-axis to locate points (x, y).
    • 2Distance Formula: Length = √[(x₂ - x₁)² + (y₂ - y₁)²], derived from Pythagoras' theorem.
    • 3Midpoint Formula: M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 ), which finds the average of the coordinates.
    • 4Gradient Formula: m = (y₂ - y₁)/(x₂ - x₁), representing the steepness or 'rise over run'.
    • 5Equation of a Line: The standard form is y = mx + c, where 'm' is the gradient and 'c' is the y-intercept.
    • 6Parallel lines have the same gradient (m₁ = m₂).
    • 7Perpendicular lines have gradients whose product is -1 (m₁ × m₂ = -1).

    Pakistan Example

    Motorway M2 Route Planning

    Imagine using a coordinate grid to map the Motorway M2. Let Lahore be at coordinate L(10, 20) and Islamabad be at I(370, 180). An engineer can use coordinate geometry to solve practical problems: 1. **Distance:** Calculate the straight-line distance between the two cities on the map using the distance formula to estimate material needs. 2. **Midpoint:** Find the exact coordinates for a service station to be built halfway between Lahore and Islamabad using the midpoint formula. 3. **Gradient:** If a new road from a town at T(200, 20) must connect to the motorway at a perpendicular angle, the engineer would first find the gradient of the M2 (line LI), and then use the perpendicular gradient rule (m₁ × m₂ = -1) to find the gradient of the new connecting road.

    Quick Revision Infographic

    Mathematics — Quick Revision

    Coordinate Geometry

    Key Concepts

    1The Cartesian plane uses a horizontal x-axis and vertical y-axis to locate points (x, y).
    2Distance Formula: Length = √[(x₂ - x₁)² + (y₂ - y₁)²], derived from Pythagoras' theorem.
    3Midpoint Formula: M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 ), which finds the average of the coordinates.
    4Gradient Formula: m = (y₂ - y₁)/(x₂ - x₁), representing the steepness or 'rise over run'.
    5Equation of a Line: The standard form is y = mx + c, where 'm' is the gradient and 'c' is the y-intercept.
    6Parallel lines have the same gradient (m₁ = m₂).

    Formulas to Know

    Formula: Length = √[(x₂ - x₁)² + (y₂ - y₁)²], derived from Pythagoras' theorem.
    Formula: M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 ), which finds the average of the coordinates.
    Formula: m = (y₂ - y₁)/(x₂ - x₁), representing the steepness or 'rise over run'.
    Line: The standard form is y = mx + c, where 'm' is the gradient and 'c' is the y-intercept.
    Pakistan Example

    Motorway M2 Route Planning

    Imagine using a coordinate grid to map the Motorway M2. Let Lahore be at coordinate L(10, 20) and Islamabad be at I(370, 180). An engineer can use coordinate geometry to solve practical problems: 1. **Distance:** Calculate the straight-line distance between the two cities on the map using the distance formula to estimate material needs. 2. **Midpoint:** Find the exact coordinates for a service station to be built halfway between Lahore and Islamabad using the midpoint formula. 3. **Gradient:** If a new road from a town at T(200, 20) must connect to the motorway at a perpendicular angle, the engineer would first find the gradient of the M2 (line LI), and then use the perpendicular gradient rule (m₁ × m₂ = -1) to find the gradient of the new connecting road.

    SeekhoAsaan.com — Free RevisionCoordinate Geometry Infographic

    Test Your Knowledge!

    5 questions to test your understanding.

    Start Quiz
    Trigonometry & BearingsVectors & Transformations