Unit 0 · Foundations of Geometry

Lesson 0.3

Distance and Midpoint on the Coordinate Plane

Quick Answer

The distance between two points on the coordinate plane can be found using horizontal or vertical distance or the Distance Formula. The midpoint is found by averaging the x-coordinates and averaging the y-coordinates.

Why this matters: Distance and midpoint are used to measure segments, locate centers, analyze figures, and solve coordinate-geometry problems.

Distance and midpoint are used to measure segments, locate centers, analyze figures, and solve coordinate-geometry problems.

Essential Question

How can coordinates be used to find the distance and midpoint between two points?

Learning Goals

1Find horizontal distance between two points.
2Find vertical distance between two points.
3Use the Distance Formula.
4Use the Midpoint Formula.
5Identify the midpoint of a segment on the coordinate plane.
6Find a missing endpoint in simple midpoint problems.
7Check whether a coordinate answer is reasonable from the graph.

Key Vocabulary

Coordinate plane

A plane formed by a horizontal x-axis and a vertical y-axis.

Ordered pair

A pair of numbers written as (x, y) that locates a point.

x-coordinate

The first coordinate in an ordered pair, showing horizontal position.

y-coordinate

The second coordinate in an ordered pair, showing vertical position.

Horizontal distance

The distance between two points with the same y-coordinate.

Vertical distance

The distance between two points with the same x-coordinate.

Distance Formula

A formula used to find the distance between any two points on the coordinate plane.

Midpoint

The point exactly halfway between two endpoints.

Midpoint Formula

A formula that averages the x-coordinates and y-coordinates of two endpoints.

Endpoint

A point at either end of a segment.

Key Ideas

Coordinates allow us to assign numbers to geometric points, turning distance and midpoint problems into arithmetic. This lesson connects the coordinate plane to segment measurement.

A. Reading Ordered Pairs

Coordinates are written (x, y) — x first, y second.
Move horizontally first (left or right), then vertically (up or down).
A negative x-coordinate means the point is to the left of the y-axis.
A negative y-coordinate means the point is below the x-axis.
The origin is (0, 0) — the intersection of both axes.

Coordinate plane with four plotted points — one in each quadrant.

B. Horizontal Distance

When two points share the same y-coordinate, the segment between them is horizontal.
Horizontal distance = |x₂ − x₁|
Only the x-coordinates are used — the y-coordinate cancels out.

Horizontal segment: both points share y = 3. Distance = |x₂ − x₁|.

C. Vertical Distance

When two points share the same x-coordinate, the segment between them is vertical.
Vertical distance = |y₂ − y₁|
Only the y-coordinates are used — the x-coordinate cancels out.

Vertical segment: both points share x = 2. Distance = |y₂ − y₁|.

D. The Distance Formula

For any two points (x₁, y₁) and (x₂, y₂):
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
This formula comes from the Pythagorean Theorem applied to the horizontal and vertical legs of a right triangle.
The horizontal leg has length |x₂ − x₁| and the vertical leg has length |y₂ − y₁|.

The Distance Formula comes from the Pythagorean Theorem: d² = (Δx)² + (Δy)².

E. Substituting Coordinates Correctly

Match x-values with x-values and y-values with y-values.
Use parentheses around negative coordinates: (x₂ − (−3)) = (x₂ + 3).
Square both differences before adding.
Simplify under the radical carefully before taking the square root.
The order of subtraction does not matter because both differences are squared.

F. Exact and Approximate Distance

An exact answer may contain a radical: d = √50 = 5√2.
An approximate answer uses a decimal: d ≈ 7.07.
Do not round until the final step — rounding early introduces error.
Include units when units are given in the problem.
Simplify radicals when possible: √50 = √(25 · 2) = 5√2.

G. The Midpoint Formula

The midpoint M of a segment with endpoints (x₁, y₁) and (x₂, y₂) is:
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Each coordinate of the midpoint is the average of the corresponding endpoint coordinates.
The midpoint lies exactly halfway between the two endpoints.

Midpoint M is equidistant from both endpoints — tick marks confirm equal halves.

H. Finding a Missing Endpoint

Write the Midpoint Formula for x: (x₁ + x₂)/2 = Mₓ
Substitute the known x-coordinate and the midpoint x-coordinate.
Solve for the missing x-coordinate.
Repeat for y: (y₁ + y₂)/2 = M_y
Solve for the missing y-coordinate.
Check: average the two endpoints and confirm you get the stated midpoint.

A is the known endpoint; M is the midpoint; B is the unknown endpoint.

Worked Examples

Example 1 — Horizontal Distance

Problem: Find the distance between A(−4, 3) and B(5, 3).

Horizontal segment: both points share y = 3. Distance = |x₂ − x₁|.

Step 1: Both points have y = 3, so the segment is horizontal.

Step 2: Use horizontal distance: AB = |x₂ − x₁|

Step 3: Substitute: AB = |5 − (−4)| = |5 + 4| = |9|

AB = 9

Check: counting from −4 to 5 on the x-axis gives 9 units. ✓

Example 2 — Vertical Distance

Problem: Find the distance between C(2, −5) and D(2, 4).

Vertical segment: both points share x = 2. Distance = |y₂ − y₁|.

Step 1: Both points have x = 2, so the segment is vertical.

Step 2: Use vertical distance: CD = |y₂ − y₁|

Step 3: Substitute: CD = |4 − (−5)| = |4 + 5| = |9|

CD = 9

Check: counting from −5 to 4 on the y-axis gives 9 units. ✓

Example 3 — Distance Formula

Problem: Find the distance between E(1, 2) and F(4, 6).

The Distance Formula comes from the Pythagorean Theorem: d² = (Δx)² + (Δy)².

Step 1: Write the Distance Formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Step 2: Substitute: d = √[(4 − 1)² + (6 − 2)²]

Step 3: Simplify: d = √[3² + 4²] = √[9 + 16] = √25

d = 5

Check: 3-4-5 right triangle. ✓

Example 4 — Midpoint Formula

Problem: Find the midpoint of the segment with endpoints G(−2, 4) and H(6, −2).

Midpoint M is equidistant from both endpoints — tick marks confirm equal halves.

Step 1: Write the Midpoint Formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Step 2: Substitute x-coordinates: (−2 + 6)/2 = 4/2 = 2

Step 3: Substitute y-coordinates: (4 + (−2))/2 = 2/2 = 1

M = (2, 1)

Check: (2, 1) lies halfway between (−2, 4) and (6, −2). ✓

Example 5 — Missing Endpoint

Problem: Endpoint A(2, 3) and midpoint M(5, 7). Find endpoint B.

A is the known endpoint; M is the midpoint; B is the unknown endpoint.

Step 1 (x): (2 + x₂)/2 = 5 → 2 + x₂ = 10 → x₂ = 8

Step 2 (y): (3 + y₂)/2 = 7 → 3 + y₂ = 14 → y₂ = 11

B = (8, 11)

Check: midpoint of (2,3) and (8,11) = ((2+8)/2, (3+11)/2) = (5, 7) ✓

Multiple Choice Check

Choose the best answer. Answers are in the Answer Key tab.

Question 1

Which ordered pair correctly describes point P plotted 3 units to the left of the y-axis and 4 units above the x-axis?

Coordinate plane with four plotted points — one in each quadrant.

A(3, 4)
B(4, −3)
C(−3, 4)
D(−4, 3)

Question 2

Points J(−3, 5) and K(7, 5) lie on a horizontal segment. What is JK?

Horizontal segment: both points share y = 5. Distance = |x₂ − x₁|.

Question 3

Use the Distance Formula. What is the distance between P(0, 0) and Q(6, 8)?

The Distance Formula comes from the Pythagorean Theorem: d² = (Δx)² + (Δy)².

A7
B10
C√28
D14

Question 4

What is the midpoint of the segment with endpoints R(−4, 2) and S(8, 6)?

Midpoint M is equidistant from both endpoints — tick marks confirm equal halves.

A(4, 8)
B(2, 4)
C(−2, −4)
D(6, 4)

Question 5

Endpoint A is at (1, 3) and midpoint M is at (4, 7). What is endpoint B?

A is the known endpoint; M is the midpoint; B is the unknown endpoint.

A(7, 11)
B(2.5, 5)
C(5, 10)
D(3, 4)

Guided Practice

Work through each problem. Reveal the solution when ready.

Find the distance between A(−6, 2) and B(4, 2).

Find the distance between C(3, −4) and D(3, 9).

Use the Distance Formula to find the distance between E(−1, 1) and F(2, 5).

Find the midpoint of the segment with endpoints G(2, −6) and H(8, 4).

Endpoint A is at (−3, 1) and midpoint M is at (2, 5). Find endpoint B.

⚠️

Common Mistakes

❌

Reversing x and y coordinates — writing (y, x) instead of (x, y)

✅

Coordinates are always written (x, y). The x-coordinate comes first and shows horizontal position.

❌

Pairing an x-coordinate with a y-coordinate in the Distance Formula

✅

Subtract x-values from x-values and y-values from y-values: (x₂ − x₁) and (y₂ − y₁) separately.

❌

Dropping the negative sign when substituting negative coordinates

✅

Use parentheses: (x₂ − (−3)) = (x₂ + 3). Skipping parentheses changes the sign.

❌

Forgetting to square both differences in the Distance Formula

✅

Both (x₂ − x₁) and (y₂ − y₁) must be squared before adding under the radical.

❌

Adding coordinates instead of averaging them for the midpoint

✅

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

❌

Averaging an x-coordinate with a y-coordinate for the midpoint

✅

Average the two x-values together and the two y-values together — never mix axes.

❌

Rounding too early in a multi-step calculation

✅

Keep the exact radical or full decimal until the final step to avoid accumulated rounding error.

❌

Finding the missing endpoint but not verifying the answer

✅

Substitute both endpoints back into the Midpoint Formula and confirm you get the stated midpoint.

Before You Finish

Make sure you can answer yes to each of these:

✓Horizontal distance uses only x-coordinate differences.
✓Vertical distance uses only y-coordinate differences.
✓General distance uses the Distance Formula: d = √[(x₂−x₁)²+(y₂−y₁)²].
✓Midpoint coordinates are averages: M = ((x₁+x₂)/2, (y₁+y₂)/2).
✓Negative coordinates require careful parentheses.
✓Exact answers may contain radicals; approximate answers use decimals.
✓A midpoint must lie exactly halfway between both endpoints.

Check Your Understanding

1. Why must x-values be paired with x-values in the Distance Formula?

2. How is the Midpoint Formula different from the Distance Formula?

3. How can you check a missing endpoint answer?

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