A8: Coordinates

Foundation Higher AQAEdexcelOCREduqasCCEA

Work with coordinates in all four quadrants

Exam Board:

📋 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)

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)

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

🧮 Worked Examples — Exam Style

Coordinates: x first (across), y second (up). Midpoint = average the x’s, average the y’s.

Find the midpoint of (2, −3) and (8, 5).

x: (2 + 8) ÷ 2 = 5 · y: (−3 + 5) ÷ 2 = 1

Midpoint: (5, 1)

M1 for averaging, A1 for (5, 1). Works with negatives: the average of −3 and 5 is +1.

Write down the coordinates of the point 4 units LEFT and 2 units DOWN from (1, 6).

Left = x decreases: 1 − 4 = −3. Down = y decreases: 6 − 2 = 4

(−3, 4)

B1 for (−3, 4). Left/right change x; up/down change y — never both.

Exam wording to expect: "Write down the coordinates of …", "Find the midpoint of A and B", "Plot the point …".

❓ 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)?

Q7: Find the midpoint of (−4, 7) and (2, −1)

Q8: A square has vertices (1,1), (5,1) and (5,5). Write the fourth vertex.

✅ Answers

2nd quadrant
(1, 5)
(7, 5)
(2, -1)
(3, 0)
6 units

📋 Model Answers — Full Working

Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.

Q1. Negative x, positive y → 2nd quadrant

Q2. 3 right from (−2, 5): −2 + 3 → (1, 5)

Q3. x: (4 + 10)/2 = 7; y: (2 + 8)/2 = 5 → (7, 5)

Q4. x: (−2 + 6)/2 = 2; y: (3 + (−5))/2 = −1 → (2, −1)

Q5. Directly below on the x-axis → same x, y = 0 → (3, 0)

Q6. Same x (vertical line): 5 − (−1) = 6 units

Q7. x: (−4 + 2)/2 = −1; y: (7 + (−1))/2 = 3 → (−1, 3)

Q8. Missing corner completes the square: (1, 5)

📌 Key Facts to Memorise

(x, y): across first, up second · origin (0, 0)
Quadrants: 1 (+,+) · 2 (−,+) · 3 (−,−) · 4 (+,−) — anticlockwise from top-right
Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
Same x-coordinate → vertical line; same y → horizontal line
Vertical distance = difference in y; horizontal distance = difference in x

🗺️ In the Exam

Coordinates feed straight-line graphs, transformations and Pythagoras distance: 1–3 marks each time. Midpoint and quadrant recall are Paper 1 regulars.

Links to: straight-line graphs (A9/A10), transformations (G7), Pythagoras (G20).

⚡ Targeted Drills

Cover the answers, work each one, then mark yourself. If any takes over a minute, revisit the matching section above.

Midpoint of (2, 8) and (6, 2) → (4, 5)
Quadrant of (−7, −1) → 3rd · of (5, −2) → 4th
Point 5 left and 3 up from (4, 1) → (−1, 4)
Midpoint of (−6, 3) and (2, −5) → (−2, −1)
Distance between (1, 2) and (1, 9) → 7 units (vertical)

🧭 Board Focus — AQA

Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.

Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.

🎯 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)

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.

📝 Exam Questions by Topic

AQA Past Papers - Topic Questions -- AQA
Edexcel Past Papers - Topic Questions -- Edexcel
OCR Past Papers - Topic Questions -- OCR
Eduqas Past Papers - Topic Questions -- Eduqas
CCEA Past Papers - Topic Questions -- CCEA

A8: Coordinates

📋 Key Concepts

📝 The Four Quadrants

📝 Plotting Coordinates

📝 Finding Midpoints

📝 Special Lines

📝 Distance Between Points

🧮 Worked Examples — Exam Style

❓ Practice Questions

✅ Answers

📋 Model Answers — Full Working

📌 Key Facts to Memorise

🗺️ In the Exam

⚡ Targeted Drills

🧭 Board Focus — AQA

🎯 Exam Tips

🧠 Problem-Solving Strategies

Problem-Solving

⚠️ Common Errors

Watch Out!

✍️ 6-Mark Exam Question

Extended Answer

📊 AO3: Reason & Interpret

Reasoning and Interpretation

📝 Exam Questions by Topic

🎬 Video Resources

Ready to ace your GCSE Mathematics exams?

🧠 Flashcards (Spaced Repetition)

📝 Exam Questions by Topic

🎯 Target Tests (Auto-Graded)

🎓 Smart Lesson (Guided)

A8: Coordinates

📋 Key Concepts

📝 The Four Quadrants

📝 Plotting Coordinates

📝 Finding Midpoints

📝 Special Lines

📝 Distance Between Points

🧮 Worked Examples — Exam Style

❓ Practice Questions

✅ Answers

📋 Model Answers — Full Working

📌 Key Facts to Memorise

🗺️ In the Exam

⚡ Targeted Drills

🧭 Board Focus — AQA

🎯 Exam Tips

🧠 Problem-Solving Strategies

Problem-Solving

⚠️ Common Errors

Watch Out!

✍️ 6-Mark Exam Question

Extended Answer

📊 AO3: Reason & Interpret

Reasoning and Interpretation

📝 Exam Questions by Topic

🎬 Video Resources

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Ready to ace your GCSE Mathematics exams?

🧠 Flashcards (Spaced Repetition)

📝 Exam Questions by Topic

🎯 Target Tests (Auto-Graded)

🎓 Smart Lesson (Guided)