What is the Distance Between Two Points on a Grid?

S1-SA3-0446

Grade Level:

Class 3

All STEM domains, Finance, Economics, Data Science, AI, Physics, Chemistry

Definition

What is it?

The distance between two points on a grid tells us how far apart those two points are. We find this by counting the number of steps horizontally and vertically, and then using a special rule (the Pythagorean theorem) to find the straight line distance.

Simple Example

Quick Example

Imagine your school map is a grid. If your classroom is at point (2,3) and the library is at (6,6), finding the distance between them means figuring out how many steps you need to take to go from your classroom to the library in a straight line.

Worked Example

Step-by-Step

Let's find the distance between Point A (1,2) and Point B (5,5) on a grid.

1. First, find the horizontal distance (change in x-coordinates). Subtract the x-values: 5 - 1 = 4 units.
---2. Next, find the vertical distance (change in y-coordinates). Subtract the y-values: 5 - 2 = 3 units.
---3. Now, imagine a right-angled triangle. The horizontal distance is one side (4), and the vertical distance is the other side (3).
---4. To find the straight-line distance (the hypotenuse), we use the Pythagorean theorem: distance^2 = (horizontal distance)^2 + (vertical distance)^2.
---5. So, distance^2 = 4^2 + 3^2 = 16 + 9 = 25.
---6. To find the distance, take the square root of 25. distance = sqrt(25) = 5 units.
---Answer: The distance between Point A (1,2) and Point B (5,5) is 5 units.

Why It Matters

Understanding distance on a grid is super important for many fields. Urban planners use it to design cities, app developers use it to show you the shortest route on Google Maps, and engineers use it to build bridges and buildings safely. It's key for careers in technology, science, and even logistics for delivery services like Zepto!

Common Mistakes

MISTAKE: Simply adding the x and y coordinates of the two points. | CORRECTION: You need to find the *difference* between the x-coordinates and the *difference* between the y-coordinates, then use the Pythagorean theorem.

MISTAKE: Forgetting to take the square root at the end. | CORRECTION: Remember that after calculating (change in x)^2 + (change in y)^2, you have distance^2. You must take the square root to get the actual distance.

MISTAKE: Mixing up x and y coordinates when subtracting. | CORRECTION: Always subtract the x-coordinates from each other to get the horizontal distance, and the y-coordinates from each other to get the vertical distance.

Practice Questions

Try It Yourself

QUESTION: What is the distance between Point P (2,1) and Point Q (5,1)? | ANSWER: 3 units

QUESTION: Find the distance between Point M (1,3) and Point N (4,7). | ANSWER: 5 units

QUESTION: A robot starts at (0,0) and moves to (3,4). Then it moves from (3,4) to (7,7). What is the total distance the robot traveled? | ANSWER: 5 + 5 = 10 units

MCQ

Quick Quiz

What is the distance between (0,0) and (3,4) on a grid?

7 units

5 units

12 units

1 unit

The Correct Answer Is:

The horizontal distance is 3-0 = 3, and the vertical distance is 4-0 = 4. Using the Pythagorean theorem, distance^2 = 3^2 + 4^2 = 9 + 16 = 25. So, the distance is sqrt(25) = 5 units.

Real World Connection

In the Real World

When you order food on Swiggy or Zomato, the app calculates the shortest distance between the restaurant and your home using this very concept. It helps delivery riders plan their routes efficiently, ensuring your delicious meal arrives hot and fast!

Key Vocabulary

Key Terms

COORDINATES: A set of numbers (x, y) that show the exact position of a point on a grid. | GRID: A network of intersecting lines forming squares, used to locate points. | HORIZONTAL DISTANCE: The difference between the x-coordinates of two points. | VERTICAL DISTANCE: The difference between the y-coordinates of two points. | PYTHAGOREAN THEOREM: A rule (a^2 + b^2 = c^2) used to find the sides of a right-angled triangle.

What's Next

What to Learn Next

Great job learning about distance on a grid! Next, you can explore how to find the midpoint between two points, which helps you locate the exact middle spot. This builds on your understanding of coordinates and distances.