Analytical Geometry: Lines

Master straight lines in the coordinate plane

CAPS Grade 10 Mathematics

Interactive Line Graph: y = mx + c

m (gradient)

c (y-intercept)

Current Line: y = 2x + 1

Hover over graph: x = 0, y = 1

This topic covers straight lines in the coordinate plane, including distance, midpoint, gradient, and equations. Use the interactive graph above - change m (gradient) and c (y-intercept) and click Update to see how the line changes!

Learning Outcomes

Plot points and navigate the Cartesian coordinate system
Calculate distance between points and find midpoints
Determine gradient (slope) of lines
Write equations of straight lines in different forms
Identify parallel and perpendicular lines
Determine if points are collinear

The Cartesian Coordinate System

Coordinate Plane

x-axis: Horizontal axis
y-axis: Vertical axis
Origin: (0,0)
Quadrants: I (+,+), II (-,+), III (-,-), IV (+,-)

Plotting Points

Ordered pair: (x, y)
x-coordinate: Horizontal position
y-coordinate: Vertical position
Example: (3, -2) → right 3, down 2

Example Points

(2,3) → Quadrant I

(-1,4) → Quadrant II

(-3,-2) → Quadrant III

(4,-1) → Quadrant IV

Quiz 1 - Coordinate System

Which quadrant contains the point (-3, 5)?

A) Quadrant I

B) Quadrant II

C) Quadrant III

D) Quadrant IV

Distance Between Two Points

Distance Formula
d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Example: Finding Distance

Problem

Find distance between (1,2) and (4,6)

Solution

d = √[(4-1)² + (6-2)²] = √[3² + 4²] = √[9 + 16] = √25 = 5 units

Quiz 2 - Distance

Distance between (0,0) and (3,4)?

A) 5

B) 7

C) 25

D) 1

Midpoint of a Line Segment

Midpoint Formula
M = [(x₁ + x₂)/2, (y₁ + y₂)/2]

Example 1

(3,5) and (7,1) → M = (5,3)

Example 2

(-4,2) and (0,-6) → M = (-2,-2)

Quiz 3 - Midpoint

Midpoint of (2,4) and (6,8)?

A) (3,5)

B) (4,6)

C) (5,7)

D) (8,12)

Gradient (Slope) of a Line

Gradient Formula
m = (y₂ - y₁) / (x₂ - x₁)

Positive Gradient

Line rises left to right

Example: m = 2

Negative Gradient

Line falls left to right

Example: m = -1.5

Zero & Undefined

Zero: horizontal (m=0)
Undefined: vertical line

Quiz 4 - Gradient

Gradient through (1,3) and (4,9)?

A) 1

B) 2

C) 3

D) 4

Equation of a Straight Line

Slope-Intercept Form

y = mx + c

m = gradient, c = y-intercept

Point-Slope Form

y - y₁ = m(x - x₁)

Use with point and gradient

Finding Equations

Examples

Gradient 2, y-intercept 3 → y = 2x + 3
Through (1,5), gradient -1 → y = -x + 6
Through (2,3) and (4,7) → y = 2x - 1

Parallel and Perpendicular Lines

∥

Parallel Lines

Equal gradients: m₁ = m₂

Example: y=2x+1, y=2x-3

⟂

Perpendicular Lines

Negative reciprocals: m₁ × m₂ = -1

Example: m₁=3, m₂=-1/3

Quiz 5 - Line Relationships

Are y = 3x + 2 and y = 3x - 4 parallel?

A) Yes

B) No

C) Perpendicular

Collinear Points

Definition: Three or more points are collinear if they lie on the same straight line.

Example 1 (Collinear)

(1,2), (3,6), (5,10)

m₁ = 2, m₂ = 2 → Equal → Collinear ✓

Example 2 (Not Collinear)

(-2,-1), (0,1), (2,4)

m₁ = 1, m₂ = 1.5 → Not equal → Not collinear ✗

Practice & Assess

Match - Formulas

√[(x₂-x₁)²+(y₂-y₁)²]

Distance

[(x₁+x₂)/2, (y₁+y₂)/2]

Midpoint

(y₂-y₁)/(x₂-x₁)

Gradient

y = mx + c

Line equation

Fill - Distance Formula

d = √[(x₂ - x₁)² + (___ - y₁)²]

Practice Questions

Distance between (-1,4) and (3,-2)

Answer: √52 = 2√13 units

Midpoint of (5,-3) and (-1,7)

Answer: (2, 2)

Gradient through (-2,5) and (4,-1)

Answer: m = -1

Summary

Distance: d = √[(x₂-x₁)² + (y₂-y₁)²]

Midpoint: M = [(x₁+x₂)/2, (y₁+y₂)/2]

Gradient: m = (y₂-y₁)/(x₂-x₁)

Line Equation: y = mx + c or y - y₁ = m(x - x₁)

Parallel: m₁ = m₂, Perpendicular: m₁ × m₂ = -1

Key Terms

Cartesian PlaneQuadrantOriginDistanceMidpointGradientSlopeY-interceptParallelPerpendicularCollinear

Circles Trigonometry