Coordinate Distance — Find distance along horizontal or...

SECTION 1

Historical Context & Motivation

Long before GPS or digital maps, people needed a way to describe exactly where something was located. Ancient civilizations used landmarks and rough measurements. But as cities grew and trade routes expanded, people needed a more exact system.

The idea of pinpointing a location with numbers came from a French mathematician named René Descartes. Legend says he was lying in bed watching a fly crawl across the ceiling. He realized he could describe the fly's exact position using two numbers — one for how far left or right, and one for how far up or down.

~300 BC

Euclid's Geometry

The Greek mathematician Euclid wrote about distance and shapes, but he had no number-grid system to place them on.

1637

Descartes Invents the Coordinate Plane

René Descartes published his idea of using two number lines (axes) to create a grid. This Cartesian plane let people describe any point with an (x, y) pair.

1700s

Coordinate Geometry Grows

Mathematicians began using coordinates to measure distances, draw graphs, and solve real-world problems in science and engineering.

Today

GPS & Modern Maps

Every phone and car GPS uses coordinate systems. Finding distance between coordinates is the foundation of digital navigation.

On the SHSAT, you will see questions that ask: How far apart are two points? When those two points line up horizontally or vertically, there is a simple shortcut. That shortcut is what this lesson is all about.

SECTION 2

Core Principles & Definitions

Before we jump into finding distances, let's review a few key ideas. These are the building blocks you need.

The Coordinate Plane

A flat grid made by two number lines crossing at a point called the origin (0, 0). The horizontal line is the x-axis and the vertical line is the y-axis.

Ordered Pairs

Every point is written as (x, y). The first number tells how far left or right. The second tells how far up or down.

Horizontal Distance

When two points share the same y-coordinate, they sit on the same horizontal line. The distance between them depends only on how their x-coordinates differ.

Vertical Distance

When two points share the same x-coordinate, they sit on the same vertical line. The distance between them depends only on how their y-coordinates differ.

Absolute Value

Distance is always positive. We use absolute value (the distance a number is from zero) to make sure our answer is never negative.

✦ KEY TAKEAWAY

Think of the coordinate plane like a city with perfectly straight streets. If two friends live on the same street (same row or same column), finding the distance between their houses is easy — just count the blocks between them. That's exactly what horizontal and vertical distance is: counting the space between two points that share a row (same y) or a column (same x).

SECTION 3

Visual Explanation

Let's see what horizontal and vertical distances look like on an actual coordinate plane. Study the diagram below carefully.

Points A and B (amber) share the same y-coordinate (2), so the distance between them is purely horizontal. Points C and D (violet) share the same x-coordinate (3), so the distance between them is purely vertical.

Notice something important in the diagram. Points A and B are on the same horizontal line because they both have a y-coordinate of 2. The only thing that changes is the x-coordinate. Points C and D are on the same vertical line because they both have an x-coordinate of 3. The only thing that changes is the y-coordinate.

In both cases, the distance is 6 units. We find this by looking at the difference between the coordinates that change.

SECTION 4

The Math Behind It

There are two simple formulas you need. One is for horizontal distance and the other is for vertical distance. Both use absolute value (written with | | bars) to make sure the answer is always a positive number. Distance can never be negative!

HORIZONTAL DISTANCE

d = |x₂ − x₁|

Use this when the two points have the same y-coordinate. Here, x₁ and x₂ are the x-coordinates of the two points. The absolute value bars ensure the result is positive.

VERTICAL DISTANCE

d = |y₂ − y₁|

Use this when the two points have the same x-coordinate. Here, y₁ and y₂ are the y-coordinates of the two points. Again, absolute value keeps the answer positive.

💡 Why Absolute Value?

Imagine you walk from floor 3 to floor −2 in a building. You moved 5 floors, not −5 floors. If you just subtract 3 − (−2) you get 5, which works. But if you subtract (−2) − 3 you get −5. That's why we wrap the subtraction in absolute value bars: |−5| = 5. Either order of subtraction gives the correct positive distance.

ABSOLUTE VALUE REMINDER

|a| = a if a ≥ 0, |a| = −a if a < 0

The absolute value of any number is its distance from 0 on the number line. For example, |−7| = 7 and |7| = 7.

SECTION 5

Different Cases You'll See

On the SHSAT, horizontal and vertical distance problems can appear in different forms. Sometimes both points are in the same quadrant. Sometimes they are on opposite sides of an axis. Let's look at the main cases.

Four common cases: Case 1 — both coordinates positive. Case 2 — points on opposite sides of an axis (involves a negative number). Case 3 — both coordinates negative. Case 4 — one point sits on an axis (has a 0 coordinate).

Case 2 is where students make the most mistakes. When one coordinate is negative and the other is positive, subtracting a negative is the same as adding. For example, 4 − (−3) = 4 + 3 = 7. Always watch for this on the SHSAT!

⚠️ SHSAT Trap Alert

Some students just subtract the two numbers without using absolute value. If you compute −3 − 4 = −7 and forget the absolute value, you might pick a negative answer choice. Distance is always positive. If your answer is negative, take the absolute value.

SECTION 6

Worked Example

Let's walk through a full example step by step, just like you would on the SHSAT.

Finding the Distance Between (−5, 3) and (7, 3)

Step 1 — Read the CoordinatesThe two points are (−5, 3) and (7, 3). Write out the x-coordinates and y-coordinates separately. Point 1 has x₁ = −5 and y₁ = 3. Point 2 has x₂ = 7 and y₂ = 3.

Step 2 — Check: Horizontal or Vertical?Compare the y-coordinates. Both y-coordinates are 3, so the points share the same y-value. That means they are on the same horizontal line. We use the horizontal distance formula.

Same y → Horizontal distance

Step 3 — Plug into the FormulaThe formula is d = |x₂ − x₁|. Substitute the values: d = |7 − (−5)|.

Step 4 — Simplify Inside the BarsRemember, subtracting a negative is the same as adding. So 7 − (−5) = 7 + 5 = 12. Now we have d = |12|.

Step 5 — Take the Absolute ValueSince 12 is already positive, |12| = 12.

Distance = 12 units

Finding the Distance Between (4, 6) and (4, −2)

Step 1 — Read the CoordinatesThe two points are (4, 6) and (4, −2). Point 1 has x₁ = 4 and y₁ = 6. Point 2 has x₂ = 4 and y₂ = −2.

Step 2 — Check: Horizontal or Vertical?Both x-coordinates are 4, so the points share the same x-value. They are on the same vertical line. We use the vertical distance formula.

Same x → Vertical distance

Step 3 — Plug into the FormulaThe formula is d = |y₂ − y₁|. Substitute: d = |−2 − 6|.

Step 4 — Simplify and Take Absolute Value−2 − 6 = −8. Now take the absolute value: |−8| = 8.

Distance = 8 units

SECTION 7

Tips & Common Pitfalls

Knowing the formula is great, but avoiding mistakes is just as important. Here are the most common errors students make on the SHSAT and how to beat them.

Common pitfalls and solutions for coordinate distance problems

Common Mistake	Why It Happens	How to Fix It
Getting a negative distance	Forgetting to use absolute value after subtracting	Always wrap the subtraction in \| \| bars. If your answer is negative, flip the sign.
Subtracting a negative incorrectly	Writing 5 − (−3) = 2 instead of 8	Remember: subtracting a negative = adding. Write it out: 5 − (−3) = 5 + 3 = 8.
Using the wrong coordinates	Mixing up x and y, or comparing x of one point to y of another	First check which coordinate is the same. Same y? Use x-values. Same x? Use y-values.
Trying to use this on diagonal distances	The two points don't share an x or y coordinate at all	This method only works for horizontal or vertical lines. Diagonal needs a different formula (distance formula).

🎯 QUICK CHECK STRATEGY

Before you use a formula, do a quick eyeball check. If both numbers are positive, the distance should be smaller than the bigger number. If the numbers are on opposite sides of zero (one positive, one negative), the distance should be bigger than either number alone. This is like checking both sides of a street — the total width includes the left side AND the right side.

SECTION 8

Connection to the Full Distance Formula

In this lesson, we've focused on points that share an x- or y-coordinate. But what about points that don't line up horizontally or vertically? For that, you would use the distance formula, which works for any two points.

Horizontal/vertical distance vs. the full distance formula

Feature	Horizontal/Vertical Distance (This Lesson)	Full Distance Formula (Future Topic)
When to use	Points share the same x or same y	Any two points, even diagonal
Formula	\|x₂ − x₁\| or \|y₂ − y₁\|	√((x₂ − x₁)² + (y₂ − y₁)²)
Difficulty	Simple subtraction + absolute value	Requires squaring, adding, and square roots
SHSAT frequency	Very common — appears often	Less common, but still tested

Here's the cool thing: the horizontal and vertical distance formulas are actually building blocks for the full distance formula. The full formula uses the Pythagorean theorem, which combines a horizontal leg and a vertical leg to find a diagonal distance. So by mastering this lesson, you are already halfway to understanding the full formula!

SECTION 9

Practice Problems

Try these five problems on your own before checking the answers. They go from easy to challenging.

PROBLEM 1 — CONCEPTUAL

Two points are (3, 7) and (9, 7). Without doing any math, explain why the distance between them is a horizontal distance, not a vertical distance.

PROBLEM 2 — BASIC CALCULATION

Find the distance between (−4, 5) and (−4, −3).

PROBLEM 3 — INTERMEDIATE

Point M is at (−6, −2) and Point N is at (8, −2). A third point P is at (8, 5). What is the total distance if you walk from M to N and then from N to P?

PROBLEM 4 — APPLIED

On a city map, a library is at coordinates (−3, 4) and a school is at (−3, −5). Each unit on the map represents 0.25 miles. How far apart are the library and the school in miles?

PROBLEM 5 — CRITICAL THINKING

Point A is at (2, k) and Point B is at (2, −3). The distance between them is 10. Find all possible values of k.

SUMMARY

Lesson Summary

When two points share the same y-coordinate, they sit on a horizontal line, and the distance between them is |x₂ − x₁|. When two points share the same x-coordinate, they sit on a vertical line, and the distance between them is |y₂ − y₁|. Absolute value ensures the distance is always positive.

Watch out for subtracting negative numbers — remember that subtracting a negative is the same as adding. When points are on opposite sides of an axis, the distance will be larger than either coordinate alone. These horizontal and vertical distances are the building blocks for the full distance formula you'll learn later.