How to Find the Distance Between Points on a Coordinate Plane

Finding the distance between two points on a coordinate plane is a fundamental skill in mathematics, particularly in geometry and algebra. To calculate the distance between two points, we utilize the distance formula, which is based on the Pythagorean theorem. The formula is given as:

**Distance = √((x₂ - x₁)² + (y₂ - y₁)²)**

Where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.

### Steps to Use the Distance Formula
1. **Identify the Coordinates**: Start by identifying the coordinates of the two points you want to measure. Let’s say we have points A(3, 2) and B(7, 5).
2. **Subtract the Coordinates**: Subtract the x-coordinates and the y-coordinates:
- For x-values: 7 - 3 = 4
- For y-values: 5 - 2 = 3
3. **Square the Results**: Now square these differences:
- (4)² = 16
- (3)² = 9
4. **Add the Squares**: Add the results together:
- 16 + 9 = 25
5. **Take the Square Root**: Finally, take the square root of the sum:
- √25 = 5

So, the distance between points A and B is 5 units.

### Understanding Whole Numbers and Simplified Radicals
If the distance you calculate is a whole number, like in this example, you can simply state the distance as 5. However, not all distances will be whole numbers. Sometimes, you may end up with a distance that is better expressed in simplified radical form. For instance, if the points were (1, 1) and (4, 5), you would find that:
- (4 - 1)² = 3² = 9
- (5 - 1)² = 4² = 16
- Add: 9 + 16 = 25
- The square root: √25 = 5
Thus, if you ever calculate a distance that isn’t a whole number, like √2 or √8, make sure to simplify it to its lowest terms (for example, √8 simplifies to 2√2).

### Real-World Applications
Understanding how to find distances between points is not only crucial in math but also has practical applications in various fields such as physics, engineering, and even in everyday life when calculating distances on a map or planning routes. Knowing how to accurately measure distances can enhance your problem-solving skills and help you tackle more complex issues in mathematics and beyond.

In conclusion, the distance formula is a powerful tool that allows you to find the distance between any two points on a coordinate plane. Practice with more examples to strengthen your understanding and ensure you're comfortable using the formula in different scenarios.