📐 Concept diagram

01-04 - Coordinate Geometry (2D)

Phase: 1 | Subject: 01-04 Prerequisites: 01-02-linear-equations.md, 01-03-linear-inequalities.md Next subject: 01-05-linear-functions.md

Learning Objectives

By the end of this subject, you will be able to:

1. Plot points and interpret coordinates on the Cartesian plane
2. Calculate the distance between any two points using the distance formula
3. Find the midpoint of a line segment using the midpoint formula
4. Calculate the gradient (slope) of a line from two points
5. Understand the sign and magnitude of gradient

Core Content

The Cartesian Plane

René Descartes invented this system in the 1600s. It links algebra (equations) with geometry (shapes) by using two perpendicular number lines:

• x-axis: horizontal, positive to the right
• y-axis: vertical, positive upward
• Origin: where they cross, at (0, 0)
• Quadrants: 4 regions labelled I, II, III, IV counter-clockwise from top-right

$         y
         |
    II   |   I
         |
---------+--------- x
         |
    III  |   IV
         |
$

A point is written as (x, y): - x-coordinate (horizontal position): positive = right, negative = left - y-coordinate (vertical position): positive = up, negative = down

Plotting Points

Example: Plot (3, 2), (-1, 4), (2, -3), (-2, -1)

Start at origin (0, 0): - (3, 2): right 3, up 2 - (-1, 4): left 1, up 4 - (2, -3): right 2, down 3 - (-2, -1): left 2, down 1

Distance Between Two Points

Given points A(x₁, y₁) and B(x₂, y₂), the distance d comes from Pythagoras' theorem:

$d = √[(x₂ - x₁)² + (y₂ - y₁)²]
$

Why this works: Imagine a right triangle where AB is the hypotenuse. The horizontal leg has length |x₂ - x₁| and the vertical leg has length |y₂ - y₁|. By Pythagoras: d² = (x₂ - x₁)² + (y₂ - y₁)²

Example: Distance between (2, 3) and (6, 8)

1. x₁ = 2, y₁ = 3, x₂ = 6, y₂ = 8
2. d = √[(6 - 2)² + (8 - 3)²]
3. d = √[4² + 5²]
4. d = √[16 + 25]
5. d = √41 ≈ 6.40

Midpoint Formula

The midpoint M of segment AB has coordinates that are the AVERAGE of the endpoints:

$M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
$

Example: Midpoint of (1, 4) and (7, 10)

1. x-coordinate: (1 + 7)/2 = 4
2. y-coordinate: (4 + 10)/2 = 7
3. Midpoint: (4, 7)

Check: (4, 7) is exactly halfway. Distance to (1, 4): √[(4-1)² + (7-4)²] = √[9+9] = √18. Distance to (7, 10): √[(7-4)² + (10-7)²] = √[9+9] = √18. Equal! ✓

Gradient (Slope)

⚠️ THIS IS CRITICAL — gradient is the foundation of all rate-of-change thinking. In calculus, the gradient formula (y₂-y₁)/(x₂-x₁) becomes the derivative. Understanding gradient now will make calculus much easier later.

Gradient measures how steep a line is. It's the "rise over run":

$m = (y₂ - y₁) / (x₂ - x₁)
$

Positive gradient: line goes UP from left to right (uphill) Negative gradient: line goes DOWN from left to right (downhill) Zero gradient: horizontal line (flat) Undefined gradient: vertical line (straight up)

Example: Gradient from (1, 2) to (5, 10)

1. Rise: 10 - 2 = 8
2. Run: 5 - 1 = 4
3. m = 8/4 = 2

This means for every 1 unit right, we go 2 units up.

Example: Gradient from (3, 7) to (6, 1)

1. Rise: 1 - 7 = -6
2. Run: 6 - 3 = 3
3. m = -6/3 = -2

This line goes downhill: for every 1 unit right, we go 2 units down.

Important Properties

1. Parallel lines have the SAME gradient
2. Perpendicular lines have gradients that are NEGATIVE RECIPROCALS: m₁ × m₂ = -1
3. If m₁ = 2, then m₂ = -1/2
4. If m₁ = -3/4, then m₂ = 4/3

Common pitfall — vertical lines: A vertical line has equation x = c. Its gradient is UNDEFINED (division by zero: run = 0). You cannot use the gradient formula for vertical lines — the formula gives a zero denominator. Similarly, a horizontal line has gradient 0.

Common pitfall — distance formula ordering: The distance formula uses (x₂-x₁)² and (y₂-y₁)². Since squaring eliminates the sign, the order of subtraction DOES NOT matter: (x₂-x₁)² = (x₁-x₂)². You can subtract in whichever order gives positive numbers for easier mental math.

Key Terms

• Parallel lines
• Perpendicular lines

Worked Examples

Example 1: Find the distance between (-2, 5) and (4, -1)

1. x₁ = -2, y₁ = 5, x₂ = 4, y₂ = -1
2. d = √[(4 - (-2))² + (-1 - 5)²]
3. d = √[6² + (-6)²]
4. d = √[36 + 36]
5. d = √72 = √(36 × 2) = 6√2 ≈ 8.49

Example 2: Find the midpoint of (3, -4) and (9, 8)

1. x-coordinate: (3 + 9)/2 = 6
2. y-coordinate: (-4 + 8)/2 = 2
3. Midpoint: (6, 2)

Example 3: Find the gradient of the line through (2, 3) and (5, 11), then determine if it's parallel to the line through (0, 1) and (3, 10)

1. First line: m = (11 - 3)/(5 - 2) = 8/3
2. Second line: m = (10 - 1)/(3 - 0) = 9/3 = 3
3. 8/3 ≠ 3, so the lines are NOT parallel

Quiz

Q1: What does the concept of Parallel lines primarily refer to in this subject?

A) A computational error related to Parallel lines B) The definition and application of Parallel lines C) A historical anecdote about Parallel lines D) A visual representation of Parallel lines

Correct: B)

• If you chose A: This is incorrect. Parallel lines is defined as: the definition and application of parallel lines. The other options describe different aspects that are not the primary focus.
• If you chose B: Parallel lines is defined as: the definition and application of parallel lines. The other options describe different aspects that are not the primary focus. Correct!
• If you chose C: This is incorrect. Parallel lines is defined as: the definition and application of parallel lines. The other options describe different aspects that are not the primary focus.
• If you chose D: This is incorrect. Parallel lines is defined as: the definition and application of parallel lines. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of Perpendicular lines?

A) It replaces all other methods in this domain B) It is primarily a historical notation system C) It is used only in advanced research contexts D) It is used to perpendicular lines in mathematical analysis

Correct: D)

• If you chose A: This is incorrect. Perpendicular lines serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: This is incorrect. Perpendicular lines serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose C: This is incorrect. Perpendicular lines serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose D: Perpendicular lines serves the purpose described in the correct answer. The other options misrepresent its role. Correct!

Q3: Which statement about The Cartesian Plane is TRUE?

A) The Cartesian Plane is mentioned only as a historical footnote B) The Cartesian Plane is an advanced topic beyond this subject's scope C) The Cartesian Plane is a fundamental concept covered in this subject D) The Cartesian Plane is not related to this subject

Correct: C)

• If you chose A: This is incorrect. The Cartesian Plane is a fundamental concept covered in this subject. This subject covers The Cartesian Plane as part of its core content.
• If you chose B: This is incorrect. The Cartesian Plane is a fundamental concept covered in this subject. This subject covers The Cartesian Plane as part of its core content.
• If you chose C: The Cartesian Plane is a fundamental concept covered in this subject. This subject covers The Cartesian Plane as part of its core content. Correct!
• If you chose D: This is incorrect. The Cartesian Plane is a fundamental concept covered in this subject. This subject covers The Cartesian Plane as part of its core content.

Q4: Based on the worked examples in this subject, what is the correct result?

A) A different result from a common mistake B) 3. Not parallel. C) An unrelated numerical value D) The inverse of the correct answer

Correct: B)

• If you chose A: This is incorrect. The worked examples show that the result is 3. Not parallel.. The other options represent common errors.
• If you chose B: The worked examples show that the result is 3. Not parallel.. The other options represent common errors. Correct!
• If you chose C: This is incorrect. The worked examples show that the result is 3. Not parallel.. The other options represent common errors.
• If you chose D: This is incorrect. The worked examples show that the result is 3. Not parallel.. The other options represent common errors.

Q5: How are The Cartesian Plane and Plotting Points related?

A) The Cartesian Plane is a special case of Plotting Points B) The Cartesian Plane and Plotting Points are completely unrelated topics C) The Cartesian Plane and Plotting Points are closely related concepts D) The Cartesian Plane is the inverse of Plotting Points

Correct: C)

• If you chose A: This is incorrect. Both The Cartesian Plane and Plotting Points are covered in this subject as interconnected topics.
• If you chose B: This is incorrect. Both The Cartesian Plane and Plotting Points are covered in this subject as interconnected topics.
• If you chose C: Both The Cartesian Plane and Plotting Points are covered in this subject as interconnected topics. Correct!
• If you chose D: This is incorrect. Both The Cartesian Plane and Plotting Points are covered in this subject as interconnected topics.

Q6: What is a common pitfall when working with Distance Between Two Points?

A) The main error with Distance Between Two Points is using it when it is not needed B) A common mistake is confusing Distance Between Two Points with a similar concept C) Distance Between Two Points has no common misconceptions D) Distance Between Two Points is always computed the same way in all contexts

Correct: B)

• If you chose A: This is incorrect. Students often confuse Distance Between Two Points with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: Students often confuse Distance Between Two Points with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose C: This is incorrect. Students often confuse Distance Between Two Points with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: This is incorrect. Students often confuse Distance Between Two Points with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply Midpoint Formula?

A) Midpoint Formula is not practically useful B) Apply Midpoint Formula to solve problems in this subject's domain C) Avoid Midpoint Formula unless explicitly instructed D) Use Midpoint Formula only in pure mathematics contexts

Correct: B)

• If you chose A: This is incorrect. Midpoint Formula is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: Midpoint Formula is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose C: This is incorrect. Midpoint Formula is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: This is incorrect. Midpoint Formula is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. Find the distance between (0, 0) and (3, 4)
   Click for answer

√[(3-0)² + (4-0)²] = √[9+16] = √25 = 5

1. Find the midpoint of (2, 5) and (8, 9)
   Click for answer

((2+8)/2, (5+9)/2) = (5, 7)

1. Find the gradient of the line through (1, 2) and (4, 8)
   Click for answer

(8-2)/(4-1) = 6/3 = 2

1. Find the gradient of the line through (3, 7) and (6, 1)
   Click for answer

(1-7)/(6-3) = -6/3 = -2

1. Are the lines through (0, 0), (2, 3) and (1, 1), (3, 7) parallel?
   Click for answer

First: 3/2 = 1.5. Second: (7-1)/(3-1) = 6/2 = 3. Not parallel.

1. Find the distance between (-1, -2) and (3, 4)
   Click for answer

√[(3-(-1))² + (4-(-2))²] = √[4² + 6²] = √[16+36] = √52 = 2√13 ≈ 7.21

Summary

Key takeaways:

• Coordinates (x, y) locate points on the Cartesian plane
• Distance formula comes from Pythagoras: d = √[(x₂-x₁)² + (y₂-y₁)²]
• Midpoint formula averages the coordinates: M = ((x₁+x₂)/2, (y₁+y₂)/2)
• Gradient = rise/run = (y₂-y₁)/(x₂-x₁)
• Positive gradient = uphill, negative = downhill, zero = flat, undefined = vertical
• Parallel lines have equal gradients
• Perpendicular lines have gradients that multiply to -1 (negative reciprocals)

Pitfalls

• Confusing the distance formula order. Students often worry about which point is (x₁, y₁) and which is (x₂, y₂). Since the differences are squared, the order DOES NOT matter: (x₂ - x₁)² = (x₁ - x₂)². Choose whichever order gives easier arithmetic.
• Computing gradient as run/rise instead of rise/run. Gradient is rise over run: m = (y₂ - y₁)/(x₂ - x₁). A common error is to divide the change in x by the change in y, which gives the reciprocal of the true gradient.
• Treating the midpoint as (x₁ + x₂, y₁ + y₂) without dividing by 2. The midpoint is the AVERAGE of the endpoints. After adding, you must divide each coordinate by 2.
• Forgetting that perpendicular gradients are negative reciprocals, not just reciprocals. If a line has gradient 2, the perpendicular gradient is -1/2, not 1/2. The negative sign is essential: m₁ × m₂ = -1.
• Attempting to calculate the gradient of a vertical line. A vertical line (x = c) has undefined gradient because run = 0, causing division by zero in the gradient formula. Don't try to use the gradient formula for vertical lines — recognise them directly.

Next Steps

Next up: 01-05-linear-functions.md