Lines in Coordinate Plane

📍 Lines in the Coordinate Plane: Your Map to Mathematical Roads

Imagine you’re a city planner. Every street in your city can be described with just a few numbers. Let’s learn how!

🎢 What is Slope? The “Steepness” of a Road

Think of slope like a slide at the playground. Some slides are steep (scary fast!), some are gentle (nice and slow).

The Big Idea

Slope tells you how tilted a line is. It answers: “For every step I walk forward, how much do I go up or down?”

The Formula

slope (m) = rise / run
         = (y₂ - y₁) / (x₂ - x₁)

🎯 Simple Example

You have two points: (1, 2) and (3, 6)

rise = 6 - 2 = 4 (went UP 4)
run  = 3 - 1 = 2 (went RIGHT 2)
slope = 4 / 2 = 2

What does slope = 2 mean? For every 1 step right, you go 2 steps up! 🚶‍♂️⬆️⬆️

🌟 Types of Slopes

graph TD
    A["Look at the line"] --> B{Which way?}
    B -->|Up to right| C["Positive slope +"]
    B -->|Down to right| D["Negative slope -"]
    B -->|Flat| E["Zero slope = 0"]
    B -->|Straight up/down| F["Undefined slope"]

📝 Line Equation Forms: Three Ways to Write the Same Road

Just like you can describe your house address in different ways, lines have different “addresses” too!

1️⃣ Slope-Intercept Form (Most Popular!)

y = mx + b

• m = slope (how steep)
• b = y-intercept (where line crosses y-axis)

Example: y = 2x + 3

• Slope is 2 (goes up 2 for every 1 right)
• Crosses y-axis at (0, 3)

2️⃣ Point-Slope Form (When You Know a Point)

y - y₁ = m(x - x₁)

Use this when you know:

• One point (x₁, y₁) on the line
• The slope m

Example: Line through (2, 5) with slope 3

y - 5 = 3(x - 2)
y - 5 = 3x - 6
y = 3x - 1

3️⃣ Standard Form (The Formal Version)

Ax + By = C

Where A, B, C are integers and A is positive.

Example: 2x + 3y = 12

🔄 Converting Between Forms

graph TD
    A["y = 2x + 4<br>Slope-Intercept"] --> B["y - 4 = 2x - 0<br>Point-Slope"]
    A --> C["2x - y = -4<br>or -2x + y = 4<br>Standard Form"]
    B --> A
    C --> A

🛤️ Parallel Lines: Roads That Never Meet

Big Idea: Parallel lines are like train tracks. They go in the same direction and never cross.

The Secret Rule

Parallel lines have the SAME slope!

🎯 Example

• Line 1: y = 3x + 2 → slope = 3
• Line 2: y = 3x - 5 → slope = 3

Same slope? They’re parallel! ✅

Quick Check Method

graph TD
    A["Compare slopes"] --> B{m₁ = m₂?}
    B -->|Yes| C["✅ PARALLEL"]
    B -->|No| D["❌ Not parallel"]

📍 Finding a Parallel Line

Problem: Find a line parallel to y = 2x + 1 passing through (3, 4)

Solution:

1. Same slope: m = 2
2. Use point-slope: y - 4 = 2(x - 3)
3. Simplify: y = 2x - 2

⊥ Perpendicular Lines: Roads That Cross at Right Angles

Big Idea: Perpendicular lines meet at 90 degrees (a perfect corner, like the letter L).

The Magic Rule

Perpendicular slopes are NEGATIVE RECIPROCALS!

What does that mean?

• Flip the fraction
• Change the sign

🎯 Examples

The Formula Check

m₁ × m₂ = -1 means they’re perpendicular!

Example: Are slopes 4 and -1/4 perpendicular?

4 × (-1/4) = -1 ✅ YES!

📍 Finding a Perpendicular Line

Problem: Find a line perpendicular to y = 3x + 2 passing through (6, 1)

Solution:

1. Original slope: 3
2. Perpendicular slope: -1/3
3. Use point-slope: y - 1 = -1/3(x - 6)
4. Simplify: y = -1/3x + 3

graph TD
    A["Original slope m"] --> B["Flip: 1/m"]
    B --> C["Change sign: -1/m"]
    C --> D["Perpendicular slope!"]

📏 Distance from Point to Line: The Shortest Path

Big Idea: The shortest distance from a point to a line is always a perpendicular drop (like dropping straight down from a diving board).

The Formula

For point (x₀, y₀) and line Ax + By + C = 0:

Distance = |Ax₀ + By₀ + C| / √(A² + B²)

🎯 Step-by-Step Example

Problem: Distance from point (3, 4) to line 3x + 4y - 5 = 0

Solution:

1. Identify: A=3, B=4, C=-5, x₀=3, y₀=4
2. Plug in numerator: |3(3) + 4(4) + (-5)| = |9 + 16 - 5| = |20| = 20
3. Plug in denominator: √(3² + 4²) = √(9 + 16) = √25 = 5
4. Distance = 20/5 = 4 units

🧠 Memory Trick

Think “Plug and Divide”:

• Plug the point into Ax + By + C
• Divide by the square root of A² + B²

graph TD
    A["Point x₀, y₀"] --> B["Line Ax + By + C = 0"]
    B --> C["Calculate numerator<br>|Ax₀ + By₀ + C|"]
    C --> D["Calculate denominator<br>√A² + B²"]
    D --> E["Divide = Distance!"]

📐 Angle Between Two Lines: How Sharp is the Corner?

Big Idea: When two lines cross, they form angles. We can calculate exactly how big that angle is!

The Formula

For two lines with slopes m₁ and m₂:

tan(θ) = |m₁ - m₂| / |1 + m₁×m₂|

Then θ = arctan(that result)

🎯 Example

Problem: Find the angle between lines with slopes 2 and 1/2

Solution:

tan(θ) = |2 - 0.5| / |1 + 2×0.5|
       = |1.5| / |1 + 1|
       = 1.5 / 2
       = 0.75

θ = arctan(0.75) ≈ 36.87°

⚡ Special Cases

🎯 Quick Example: Are They Perpendicular?

Lines with slopes 3 and -1/3:

1 + (3)×(-1/3) = 1 + (-1) = 0

Denominator is zero = 90° = Perpendicular! ✅

graph TD
    A["Two lines crossing"] --> B["Get both slopes m₁, m₂"]
    B --> C["Calculate tan θ formula"]
    C --> D["Use arctan to find angle"]
    D --> E["θ is your answer!"]

🎯 Quick Reference Summary

🌟 You Did It!

You now understand:

• ✅ How to measure the steepness of any line (slope)
• ✅ Three ways to write a line’s equation
• ✅ When lines are parallel (same slope)
• ✅ When lines are perpendicular (slopes multiply to -1)
• ✅ How to find the shortest distance to a line
• ✅ How to calculate the angle where lines meet

You’re now a coordinate geometry pro! 🎉