MATH 1 · CHAPTER 3

THE STRAIGHT LINE

Slope · Parallel & Perpendicular · Distance · Forms of Lines

Lines of different slopes crossing one plane

GROUP 4 — Leoder · Nagaño · Sarmiento · Mojica · Taño · Peñano · De Vera

OUR JOURNEY

Four stops, one line

1

SLOPE OF A LINE

how steep is it?

2

PARALLEL & PERPENDICULAR

how lines relate

3

DISTANCE FROM A POINT TO A LINE

the shortest path

4

FORMS OF LINES

one line, five ways to write it

+

AREA OF A TRIANGLE

application: area from coordinates alone

Then: a 10-point quiz — posted tomorrow, answered within the day.

SECTION 1 · SLOPE OF A LINE

Slope = rise over run

Line through (4,3) and (8,5) with rise 2 and run 4 marked

m = ( y₂ − y₁ ) / ( x₂ − x₁ )

m = 2 / 4 = 1/2

positive → the line RISES

SECTION 1 · WORKED EXAMPLE

Watch the steps

Points

A (4, 3) and B (8, 5)

Subtract the y's

5 − 3 = 2 → the RISE

Subtract the x's

8 − 4 = 4 → the RUN

Divide

m = 2 / 4 = 1/2

Same order matters: whichever point is “first” on top must be first at the bottom.

SECTION 1 · READING A LINE AT A GLANCE

The four faces of slope

Four panels: rising, falling, horizontal, vertical lines

Vertical line: the run is 0 — dividing by zero is impossible, so the slope is UNDEFINED.
Horizontal line: the rise is 0 — so the slope is exactly 0.

LIVE · LINE LAB

Drag the points — watch the line

Rise / run Slope m Angle θ y = mx + b General form ⊥ slope

Try: make a vertical line · a horizontal line · two points with slope −1.

SECTION 1 · ANGLE OF INCLINATION

Slope is a tangent

Line with its angle of inclination theta marked

θ = smallest positive angle from the +x axis

m = tan θ

θ acute → m positive
θ obtuse → m negative
θ = 90° → undefined

SECTION 2 · PARALLEL LINES

Parallel: same steepness, never meet

m₁ = m₂

AND different y-intercepts

Same slope + same intercept
= the SAME line, not parallel

SECTION 2 · PERPENDICULAR LINES

Perpendicular: slopes multiply to −1

Two perpendicular lines crossing at a right angle

m₁ × m₂ = −1

(−2/3) × (3/2) = −1 ✓

Flip the fraction, flip the sign:
−2/3 → 3/2

SECTION 2 · COLLINEAR POINTS

Three points, one line?

Points A, B, C lying on a single straight line

slope AB = slope BC

→ A, B, C are COLLINEAR

Close is not enough — the slopes must be EQUAL.
(Bonus test: area = 0)

SECTION 3 · DISTANCE FROM A POINT TO A LINE

Distance = the shortest path

Perpendicular dashed segment from point Q to the line

The shortest path from a point to a line is always the PERPENDICULAR one — 90°.

Any other path is longer.

Signed distance also tells you WHICH SIDE of the line you're on.

SECTION 3 · THE FORMULA AT WORK

One line, three points

d = | A x₀ + B y₀ + C | / √( A² + B² ) · for the line Ax + By + C = 0

Line: 5x − 12y + 3 = 0 → √(5² + 12²) = √169 = 13

P (−2, 1)

| −10 − 12 + 3 | / 13 = 19/13 ≈ 1.46

Q (2, −1)

| 10 + 12 + 3 | / 13 = 25/13 ≈ 1.92

R (5, 0)

| 25 − 0 + 3 | / 13 = 28/13 ≈ 2.15

The absolute value is there because distance is never negative.

SECTION 4 · FORMS OF LINES

One line, five ways to write it

TWO-POINT

y − y₁ = [(y₂−y₁)/(x₂−x₁)] (x − x₁)

POINT-SLOPE

y − y₁ = m (x − x₁)

SLOPE-INTERCEPT

y = m x + b

INTERCEPT

x/a + y/b = 1

GENERAL

A x + B y + C = 0

SECTION 4 · CHOOSING THE FORM

Match the form to the givens

Two points given	TWO-POINT FORM	through (2, 3) and (5, 9) → y = 2x − 1
Slope + one point	POINT-SLOPE FORM	m = 2 through (3, 4) → y = 2x − 2
Slope + y-intercept	SLOPE-INTERCEPT	m = −2, b = 5 → y = −2x + 5
Both intercepts	INTERCEPT FORM	2x + 5y = 20 → x/10 + y/4 = 1
Final / standard answer	GENERAL FORM	y = (2/3)x + 1 → 2x − 3y + 3 = 0

Intercept form fails for lines through the origin — one intercept would be 0.

APPLICATION · AREA OF A TRIANGLE

Area from coordinates alone

Triangle with vertices A (-4,-2), B (1,-1), C (3,4)

No base. No height.

Just the three vertices — and one determinant.

First: list the vertices COUNTERCLOCKWISE.
Order decides the sign.

APPLICATION · THE DETERMINANT WEAVE

Copy, weave, halve

Basket-weave determinant diagram: green diagonals add, red diagonals subtract

Down (add)

(−4)(−1) + (−2)(3) + (1)(4) → 4 − 6 + 4 = 2

Up (subtract)

(3)(−1) + (4)(−4) + (1)(−2) → −3 − 16 − 2 = −21

Halve the difference

Area = ½ | 2 − (−21) | = 23/2 = 11.5 square units

60-SECOND CHECK

Say these out loud

1

Slope = rise / run = tan θ. Vertical: undefined. Horizontal: 0.

2

Parallel: equal slopes, different intercepts. Perpendicular: product = −1.

3

Distance = |Ax₀ + By₀ + C| / √(A² + B²) — absolute value, always positive.

4

Five forms — pick by the givens. Triangle area: weave the determinant, then halve.

YOUR TURN

The Group 4 Quiz

10 POINTS

6 objective questions + 2 computation questions (2 pts each)
Google Forms — posted tomorrow, answered within the day

Reporters are exempt from their own quiz.

THANK YOU!

Questions?

GROUP 4 · MATH 1 · CHAPTER 3 — THE STRAIGHT LINE

WATCH & REVIEW

See it again, slower

1-minute explainer

Reference videos

▶ Khan Academy — Intro to slope

▶ Khan Academy — Finding slope from graph

Every formula in this deck is worked step by step on the Lessons page.