Coordinates and Graphs
Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.
First Principles
Objective: Understand and calculate the straight-line distance and the midpoint between any two points on a Cartesian coordinate plane.
Concrete Scenario: Imagine looking at a satellite map of Nairobi or Eldoret plotted on a grid. Point \( A(x_1, y_1) \) represents a solar-powered water borehole, and point \( B(x_2, y_2) \) is a community school. To lay a direct water pipeline or find the fair halfway meeting spot, we use the grid coordinates to calculate the exact distance and midpoint.
Geometric Insight:
- Distance: Moving from \( A \) to \( B \) creates a right-angled triangle. The horizontal leg is \( |x_2 - x_1| \) and the vertical leg is \( |y_2 - y_1| \). By Pythagoras' Theorem, the hypotenuse is the direct distance: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- Midpoint: The midpoint \( M \) is the point precisely halfway along the segment. It is simply the arithmetic mean (average) of the \( x \)-coordinates and the \( y \)-coordinates: \[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Interactive Coordinate Explorer
Drag points A and B on the grid to see the right triangle, distance, and midpoint update live.
Key Formulas
1. Distance Formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Calculates the straight-line distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \). Derived directly from Pythagoras' Theorem where \( \Delta x = x_2 - x_1 \) and \( \Delta y = y_2 - y_1 \).
2. Midpoint Formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2} \right) \]Finds the coordinates of the point equidistant from both endpoints by taking the mean of the \( x \)-coordinates and the mean of the \( y \)-coordinates.
3. Finding an Unknown Endpoint:
If the midpoint \( M(x_m, y_m) \) and one endpoint \( A(x_1, y_1) \) are known, the other endpoint \( B(x_2, y_2) \) is found using:
\[ x_2 = 2x_m - x_1, \qquad y_2 = 2y_m - y_1 \]Worked Examples
Example 1 (Easy): Distance Calculation
A delivery runner walks from point \( A(1, 2) \) to point \( B(4, 6) \). Find the direct distance between \( A \) and \( B \).
- Identify coordinates: \( x_1 = 1, y_1 = 2 \) and \( x_2 = 4, y_2 = 6 \).
- Find coordinate differences: \[ \Delta x = 4 - 1 = 3 \] \[ \Delta y = 6 - 2 = 4 \]
- Apply the distance formula: \[ d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Answer: The distance is \( 5 \) units.
Example 2 (Medium): Midpoint Calculation with Negative Coordinates
Two mobile money kiosks are positioned at \( K_1(-3, 7) \) and \( K_2(5, -1) \). Find the coordinates of the central distributor kiosk situated exactly halfway between them.
- Identify coordinates: \( (x_1, y_1) = (-3, 7) \) and \( (x_2, y_2) = (5, -1) \).
- Compute the average of the x-coordinates: \[ x_m = \frac{-3 + 5}{2} = \frac{2}{2} = 1 \]
- Compute the average of the y-coordinates: \[ y_m = \frac{7 + (-1)}{2} = \frac{6}{2} = 3 \]
Answer: The midpoint kiosk is located at \( (1, 3) \).
Example 3 (Hard): Finding an Endpoint & Total Length
A high-voltage power transmission line runs in a straight path from town \( P(-2, 4) \) through a substation \( M(3, 8) \) to town \( Q(x_2, y_2) \). If \( M \) is the exact midpoint of segment \( PQ \), find the coordinates of town \( Q \) and the total length of the power line.
- Solve for \( x_2 \): \[ \frac{-2 + x_2}{2} = 3 \implies -2 + x_2 = 6 \implies x_2 = 8 \]
- Solve for \( y_2 \): \[ \frac{4 + y_2}{2} = 8 \implies 4 + y_2 = 16 \implies y_2 = 12 \] Thus, town \( Q \) is at \( (8, 12) \).
- Calculate total length \( PQ \): \[ \Delta x = 8 - (-2) = 10, \quad \Delta y = 12 - 4 = 8 \] \[ d = \sqrt{10^2 + 8^2} = \sqrt{100 + 64} = \sqrt{164} = 2\sqrt{41} \approx 12.81 \text{ units} \]
Answer: \( Q(8, 12) \) and the total length is \( \sqrt{164} \approx 12.81 \) units.
Common Mistakes
Misconception 1: Adding differences directly instead of using Pythagoras
Misconception 2: Subtracting instead of adding coordinates for the midpoint
Real World
Practice