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What is the Distance Between Two Points in Coordinate Geometry?
Grade Level:
Class 7
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The distance between two points in coordinate geometry is the shortest length of the straight line connecting those two points. It tells us how far apart these points are on a graph using their x and y coordinates.
Simple Example
Quick Example
Imagine you have two friends, Rahul and Priya, living in different parts of your city. If Rahul's house is at (2, 3) on a map and Priya's house is at (5, 7), the distance between their houses is how far they need to travel in a straight line to meet. We use a special formula to find this exact distance.
Worked Example
Step-by-Step
Let's find the distance between point A (1, 2) and point B (4, 6).
Step 1: Identify the coordinates. Let (x1, y1) = (1, 2) and (x2, y2) = (4, 6).
---Step 2: Write down the distance formula: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2).
---Step 3: Substitute the x-values: (4 - 1)^2 = 3^2 = 9.
---Step 4: Substitute the y-values: (6 - 2)^2 = 4^2 = 16.
---Step 5: Add the results: 9 + 16 = 25.
---Step 6: Take the square root: sqrt(25) = 5.
Answer: The distance between point A (1, 2) and point B (4, 6) is 5 units.
Why It Matters
Understanding distance between points is super important for many cool fields! Engineers use it to design bridges, Computer Scientists use it for game development and graphics, and Data Scientists use it to find patterns in data. It's a foundational skill for building robots, creating GPS navigation, and even understanding how AI works.
Common Mistakes
MISTAKE: Subtracting coordinates in the wrong order, like (x1 - x2) instead of (x2 - x1) | CORRECTION: While the order of subtraction (x2-x1 or x1-x2) doesn't matter because we square the result, it's good practice to stick to (x2-x1) and (y2-y1) to avoid confusion, especially with negative numbers.
MISTAKE: Forgetting to square the differences or forgetting to take the square root at the end | CORRECTION: Remember the formula has (difference)^2 for both x and y terms, and the very last step is always taking the square root of the sum.
MISTAKE: Mixing up x and y coordinates, for example, using (x2 - y1)^2 | CORRECTION: Always subtract x-coordinates from x-coordinates and y-coordinates from y-coordinates. Keep them in their respective pairs.
Practice Questions
Try It Yourself
QUESTION: Find the distance between P (0, 0) and Q (3, 4). | ANSWER: 5 units
QUESTION: What is the distance between R (2, -1) and S (5, 3)? | ANSWER: 5 units
QUESTION: A point M is at (1, 1) and another point N is at (7, 9). Find the distance between M and N. | ANSWER: 10 units
MCQ
Quick Quiz
Which of these is the correct formula to find the distance between two points (x1, y1) and (x2, y2)?
sqrt((x2 + x1)^2 + (y2 + y1)^2)
sqrt((x2 - x1)^2 + (y2 - y1)^2)
(x2 - x1) + (y2 - y1)
(x2 - x1)^2 + (y2 - y1)^2
The Correct Answer Is:
B
Option B is the correct distance formula, also known as the Pythagorean theorem applied to coordinates. Options A, C, and D are incorrect mathematical operations for finding distance.
Real World Connection
In the Real World
Think about food delivery apps like Zomato or Swiggy. When you order food, the app calculates the distance between the restaurant and your home using this exact concept to estimate delivery time and cost. Even ISRO scientists use similar principles to track satellites and calculate distances in space!
Key Vocabulary
Key Terms
COORDINATE: A set of values (like x and y) that show an exact position on a graph | ORIGIN: The point (0, 0) where the x and y axes cross | X-AXIS: The horizontal number line on a graph | Y-AXIS: The vertical number line on a graph | PYTHAGOREAN THEOREM: A fundamental theorem in geometry that forms the basis of the distance formula.
What's Next
What to Learn Next
Great job learning about distance! Next, you can explore the 'Midpoint Formula' to find the exact middle point between two coordinates. This builds on your understanding of points and distances, helping you locate central positions.
