A8: Coordinates
Work with coordinates in all four quadrants
📋 Key Concepts
Coordinates: Pairs of numbers (x, y) that specify a position on a grid. The x-coordinate comes first, then y.
Format: (x, y)
x = horizontal position (left/right)
y = vertical position (up/down)
x = horizontal position (left/right)
y = vertical position (up/down)
Remember: "Along the corridor, up the stairs" - x first, then y.
📝 The Four Quadrants
Quadrants: The coordinate grid is divided into four sections:
| Quadrant | x | y | Examples |
|---|---|---|---|
| 1st (top right) | + | + | (3, 4), (7, 2) |
| 2nd (top left) | - | + | (-3, 4), (-5, 1) |
| 3rd (bottom left) | - | - | (-2, -5), (-4, -3) |
| 4th (bottom right) | + | - | (3, -2), (6, -4) |
Example 1
Identify the quadrant for each point:
a) (5, 3) → 1st quadrant (both positive)
b) (-2, 4) → 2nd quadrant (x negative, y positive)
c) (-3, -1) → 3rd quadrant (both negative)
d) (4, -2) → 4th quadrant (x positive, y negative)
📝 Plotting Coordinates
Example 2
Plot the points A(2, 3), B(-1, 4), C(-2, -3), D(3, -2)
Method:
A(2, 3): Start at origin, go 2 right, 3 up
B(-1, 4): Start at origin, go 1 left, 4 up
C(-2, -3): Start at origin, go 2 left, 3 down
D(3, -2): Start at origin, go 3 right, 2 down
📝 Finding Midpoints
Midpoint formula: The midpoint M of two points (x₁, y₁) and (x₂, y₂) is:
M = (x₁ + x₂⁄2, y₁ + y₂⁄2)
Example 3
Find the midpoint of A(2, 4) and B(6, 8)
Solution:
Midpoint = (2 + 6⁄2, 4 + 8⁄2)
= (8⁄2, 12⁄2)
= (4, 6)
Example 4
Find the midpoint of P(-3, 2) and Q(5, -4)
Solution:
Midpoint = (-3 + 5⁄2, 2 + (-4)⁄2)
= (2⁄2, -2⁄2)
= (1, -1)
📝 Special Lines
Key facts:
- x-axis: All points have y = 0, e.g., (3, 0), (-2, 0)
- y-axis: All points have x = 0, e.g., (0, 4), (0, -3)
- Origin: The point (0, 0) where axes meet
- Vertical line x = a: All points where x = a
- Horizontal line y = b: All points where y = b
Example 5
Find the coordinates of the point on the y-axis that is the same distance from A(2, 5) and B(2, -3).
Solution:
Points on the y-axis have x = 0, so we're looking for (0, y).
Since A and B have the same x-coordinate (2), the midpoint lies on the y-axis.
Midpoint y = 5 + (-3)⁄2 = 2⁄2 = 1
Answer: (0, 1)
📝 Distance Between Points
Horizontal distance: |x₂ - x₁| (ignore y values)
Vertical distance: |y₂ - y₁| (ignore x values)
Vertical distance: |y₂ - y₁| (ignore x values)
Example 6
Find the distance between A(2, 3) and B(7, 3)
Solution:
Since y-values are equal, distance = |7 - 2| = 5
Example 7
Find the distance between P(-2, 1) and Q(-2, -4)
Solution:
Since x-values are equal, distance = |1 - (-4)| = |1 + 4| = 5
❓ Practice Questions
Q1: In which quadrant is the point (-4, 3)?
Q2: What are the coordinates of a point 3 units right of (-2, 5)?
Q3: Find the midpoint of (4, 2) and (10, 8).
Q4: Find the midpoint of (-2, 3) and (6, -5).
Q5: Point A is (3, -2). Point B is on the x-axis, directly below A. What are the coordinates of B?
Q6: What is the distance between (-3, 5) and (-3, -1)?
✅ Answers
- 2nd quadrant
- (1, 5)
- (7, 5)
- (2, -1)
- (3, 0)
- 6 units
🎯 Exam Tips
- Always write coordinates in brackets with a comma: (x, y)
- x comes first: "along the corridor, up the stairs"
- Negative x = left of origin; negative y = below origin
- Points on x-axis have y = 0; points on y-axis have x = 0
- For midpoint, find the average of x-values and average of y-values
🧠 Problem-Solving Strategies
Problem-Solving
Use the midpoint formula to find centre points, and the distance formula (or Pythagoras) for distances. For problems involving axes, remember points on the x-axis have y = 0, and points on the y-axis have x = 0.
Multi-Step Problem
Three vertices of a parallelogram are A(1, 2), B(5, 2) and D(3, 6). Find the coordinates of vertex C.
Solution:
Step 1: Midpoint of diagonal BD = (5+3⁄2, 2+6⁄2) = (4, 4)
Step 2: Diagonals of a parallelogram bisect each other, so midpoint of AC = (4, 4)
Step 3: (1+x⁄2, 2+y⁄2) = (4, 4), so 1+x = 8 and 2+y = 8, giving C = (7, 6)
⚠️ Common Errors
Watch Out!
1. Wrong: Writing coordinates as (y, x) Correct: Always (x, y) — x comes first
2. Wrong: (-2, 5) is in the 4th quadrant Correct: (-2, 5) is in the 2nd quadrant (x negative, y positive)
3. Wrong: Midpoint formula adds coordinates and doesn't divide by 2 Correct: Midpoint = average of x-values, average of y-values — must divide by 2
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A(2, 3) and B(8, 7) are opposite vertices of a rectangle with sides parallel to the axes. (a) Find the other two vertices C and D. (b) Find the midpoint of the diagonal AB. (c) Find the area of the rectangle.
(a) The other vertices must share x and y coordinates with A and B: C = (8, 3) and D = (2, 7)
(b) Midpoint = (2+8⁄2, 3+7⁄2) = (5, 5)
(c) Length = |8 - 2| = 6, Width = |7 - 3| = 4. Area = 6 × 4 = 24 square units
Mark scheme: (a) 2 marks for both vertices. (b) 2 marks. (c) 2 marks for length, width and area.
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A phone mast at point M has signal covering all points within 5 km. Town A is at (2, 4) and Town B at (8, 4) on a map where 1 unit = 1 km.
(a) If M is at (5, 4), does Town A get signal?
(b) Does Town B get signal?
(c) Where should M be placed so both towns just get signal?
Answers: (a) Distance = |5 - 2| = 3 km < 5 km, so yes. (b) Distance = |8 - 5| = 3 km < 5 km, so yes. (c) Midpoint of A and B = (5, 4). Distance to each = 3 km < 5, so (5, 4) already works — but "just" getting signal means on the edge. Actually no single point gives both towns exactly 5 km since they're only 6 km apart. Placing M at (5, 4) gives both 3 km — well within range.
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