Coordinate Geometry and Lines

Section 1: Introduction

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Think about the last time you drove up a steep hill. Some hills feel gentle; others feel like you’re launching into orbit. That steepness — how much you rise for every metre you travel forward — has a precise mathematical name: slope.

Or think about a car’s speedometer. Speed is distance divided by time — change in position divided by change in time. That fraction, , is exactly the same idea as slope, just applied to motion instead of geometry.

Slope isn’t a geometry concept. It’s a universal concept. It measures how fast one quantity changes relative to another. That’s why it appears everywhere: in physics, in economics, in biology, in engineering.

A teaser for what’s coming. In calculus, you’ll encounter something called the derivative. It sounds intimidating — but once you’ve worked through this course, you’ll recognize it immediately: the derivative is just a slope. Specifically, it’s the slope of the curve at a single point, computed by finding slopes of secant lines through nearby points and asking what value they approach.

Everything in this lesson — distance, midpoint, slope, line equations, secant lines — is building toward that moment. You won’t learn the derivative today. But by the time you meet it in calculus, it will feel like an old friend.

After this lesson, you will be able to:

Compute the exact distance between two points using
Find the midpoint of a segment using
Compute slope as and interpret its sign and magnitude
Write equations of lines in point-slope form — the primary form used in calculus
Identify parallel and perpendicular lines using slope relationships
Find the equation of the secant line through two points on a curve

Section 2: Prerequisites

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This lesson builds directly on COM-1. The notation habits you established there — vertical work, exact values, and implication arrows — are expected throughout.

From COM-1: Exact Values. Coordinates like must stay as fractions. Do not use decimals or approximations in your work.
Cartesian Plane: You should be comfortable plotting points , identifying quadrants, and navigating the horizontal () and vertical () axes.
Fraction Arithmetic: You will need to add, subtract, multiply, and divide fractions without a calculator to compute slopes and intercepts.
Pythagorean Theorem: Recall that for a right triangle. This is the geometric engine behind the distance formula.

Retrieval Checkpoint

In the phrase “rise over run,” which direction corresponds to the “rise”?

Success Factor: Slope is the single most important concept in calculus. If you ever find yourself writing “run over rise” (), stop and re-read the definition. Getting the direction right today will save you hours of confusion in the next course.

Section 3: Core Concepts

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C1 — The Distance Formula

Place two points on a coordinate plane: and . To find the straight-line distance between them, draw a right triangle: the horizontal leg has length and the vertical leg has length . The distance is the hypotenuse.

Distance Formula

This is the Pythagorean theorem written in coordinate form. It doesn’t matter which point you call — squaring removes the sign.

The distance formula is the Pythagorean theorem in coordinate form: the legs are Δx and Δy; the hypotenuse is d.

Quick example: distance from to :

Common error — the “Manhattan distance” trap: Writing . This gives the distance you’d walk along a grid (like city blocks) — not the straight-line distance. Always square each difference, add, then take the square root.

C2 — The Midpoint Formula

The midpoint of a segment is the point exactly halfway between its two endpoints. You find it by averaging each coordinate separately.

Midpoint Formula

Add the two -values and divide by 2; add the two -values and divide by 2.

Common error — subtraction instead of addition: Writing instead of . The midpoint uses the average (sum ÷ 2), not the half-difference. If you need to, say out loud: “add, then halve.”

C3 — Slope

Slope measures steepness: how much a line rises (or falls) for every unit it moves to the right. It’s the ratio of vertical change to horizontal change.

Slope

Given any two distinct points on a line, the slope is the same regardless of which two points you choose — this is what makes slope a property of the line, not just the points.

Special cases:

Zero slope (): the line is horizontal — it doesn’t rise or fall at all
Undefined slope: the line is vertical (, so the denominator is zero) — these are written as , not

The most common slope error: reversing the formula and writing . The mnemonic is “rise over run” — rise is vertical (), run is horizontal (). The -values always go in the numerator.

C4 — Equations of Lines

Once you know a slope and one point on a line, you can write its equation. There are two standard forms — and in calculus, one of them is far more useful than the other.

Point-Slope Form (primary)

where is the slope and is any known point on the line.

Slope-Intercept Form (derived)

where is the -intercept. This is obtained by expanding point-slope and solving for .

Why point-slope is the primary form in calculus. In calculus, you’ll be given a point on a curve and a slope (the “tangent slope”) — you’ll almost never know the -intercept directly. Point-slope form lets you write the equation immediately from what you have. Starting with slope-intercept forces you to compute as an extra step — and that’s an extra opportunity for an error.

Get comfortable reaching for point-slope first. You can always expand it to slope-intercept afterwards if you need to.

Example: slope , through :

That’s it — that’s the equation. If you need slope-intercept:

C5 — Parallel and Perpendicular Lines

Parallel and Perpendicular Slope Relationships

Parallel lines have equal slopes: . They never intersect.
Perpendicular lines have slopes whose product is : , so . They intersect at a right angle.

The perpendicular slope rule is sometimes stated as “negate and take the reciprocal.” Both operations are required:

If , then (not , which only negates; not , which only reciprocates)
If , then (negate: , reciprocate: )

Common error — half the operation: Given slope 4, writing the perpendicular slope as (only negated) or (only reciprocated). The correct perpendicular slope is . Both operations are always required.

C6 — Secant Lines

A secant line is a straight line that crosses a curve at exactly two distinct points. Its slope measures the average rate of change of the curve between those two points.

Slope of a Secant Line

If a curve is defined by and the secant passes through and , then:

This is simply the slope formula — applied to two points on a curve rather than a line.

The secant line is the bridge between algebra and calculus. In the interactive visualization below, the fixed point P is anchored at on the parabola . Drag the slider to move the second point Q along the curve. Watch what happens to the slope of the secant as Q approaches P.

Move point Q: x₂ = 2.5

Slope of secant PQ: m = 3.50

What do you notice? As you drag Q toward P (sliding x₂ toward 1), the slope values get closer and closer to a specific number. That limiting value — the slope the secant approaches as the two points merge — is what calculus calls the derivative of at . You’ll formalize this in BTC-2.

Section 4: Worked Examples

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Example 1 — Distance and Midpoint (C1, C2)

Find the exact distance and midpoint between and .

Step 1 — Identify the differences:

Step 2 — Apply the distance formula:

Step 3 — Apply the midpoint formula:

Note: the midpoint -coordinate is , not 2.5 — exact form from COM-1.

Example 2 — Testing Perpendicularity with Slopes (C3, C5)

Points , , . Is the angle at a right angle? In other words, is ?

Try computing both slopes before reading the solution.

Show Solution

Step 1 — Compute slope of PQ:

Step 2 — Compute slope of PR:

Step 3 — Check the product:

Since the product of slopes is , we conclude — the angle at is exactly .

Example 3 — Equation of a Line Through Two Points (C3, C4)

Write the equation of the line through and .

Step 1 (slope) is fully shown. Try step 2 before reading the rest.

Step 1 — Compute the slope:

Show Steps 2 and 3

Step 2 — Write in point-slope form (using the point ):

Step 3 — Expand to slope-intercept if needed:

Check: Using the other point gives — the same result. Either point works in step 2.

Example 4 — The Secant Line (C3, C4, C6)

Let . Find the equation of the secant line through and .

The steps are named. Try each one before reading.

Step 1: Evaluate f at both points

Step 2: Compute the slope of the secant

Step 3: Write the equation in point-slope form

Using :

Equivalent slope-intercept form:

Notice: we evaluated at two points on a curve, computed a slope, then wrote a line equation — using nothing but C1–C4 skills. This is exactly the difference quotient from BTC-2, before taking the limit.

Section 5: Guided Practice

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Choose the correct answer from the dropdown. If your choice is wrong, you’ll see an explanation of why — and why the correct answer is right.

Problem 1 — Slope from Two Points (C3) — Generative

Find the slope of the line through and .

Problem 2 — Point-Slope Form (C4) — Generative

A line has slope and passes through . Which equation is correct?

Problem 3 — Parallel or Perpendicular? (C5)

Line 1 has slope . Line 2 has slope . What is their relationship?

Line 1 has slope . Line 2 also has slope . What is their relationship?

Line 1 has slope . Line 2 has slope . What is their relationship?

Problem 4 — Distance Formula (C1)

Find the exact distance between and .

Section 6: Independent Practice

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Work through each problem independently. Show your steps using vertical work from COM-1. Use the “Show Solution” toggle to check your reasoning — not just your answer.