Understanding the Distance Formula in Coordinate Geometry

📚 Understanding the Distance Formula

The distance formula is a crucial tool in coordinate geometry that allows us to calculate the distance between two points in a coordinate plane. It's derived from the Pythagorean theorem and provides a straightforward method for finding the length of a line segment connecting two points.

📜 A Brief History

The concept of measuring distance has been around for millennia, but the formalization of distance within a coordinate system is attributed to René Descartes, the father of analytic geometry. Descartes' work in the 17th century connected algebra and geometry, paving the way for the distance formula as we know it.

📐 Key Principles of the Distance Formula

• 📍 Definition: The distance, $d$, between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a coordinate plane is given by the formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• ➕ Derivation from Pythagorean Theorem: Imagine a right triangle where the line segment connecting the two points is the hypotenuse. The legs of the triangle are the differences in the x-coordinates ($x_2 - x_1$) and the y-coordinates ($y_2 - y_1$). The distance formula is simply applying the Pythagorean theorem ($a^2 + b^2 = c^2$) to find the length of the hypotenuse.
• 🔢 Applying the Formula: To use the formula, subtract the x-coordinates and square the result, subtract the y-coordinates and square the result, add the two squared results, and then take the square root of the sum.
• ➕ Order Doesn't Matter: Because we are squaring the differences, it doesn't matter which point you label as $(x_1, y_1)$ and which you label as $(x_2, y_2)$. The result will be the same.

🌍 Real-World Examples

Here are a couple of practical examples to illustrate how the distance formula can be applied:

1. Example 1: Find the distance between the points (1, 2) and (4, 6).
   • $d = \sqrt{(4 - 1)^2 + (6 - 2)^2}$
   • $d = \sqrt{(3)^2 + (4)^2}$
   • $d = \sqrt{9 + 16}$
   • $d = \sqrt{25}$
   • $d = 5$
2. Example 2: A map shows two cities located at (-3, 5) and (2, -1). What is the straight-line distance between the cities?
   • $d = \sqrt{(2 - (-3))^2 + (-1 - 5)^2}$
   • $d = \sqrt{(5)^2 + (-6)^2}$
   • $d = \sqrt{25 + 36}$
   • $d = \sqrt{61}$
   • $d \approx 7.81$

📝 Practice Quiz

1. Find the distance between (0, 0) and (3, 4).
2. What is the distance between (-1, 2) and (2, -2)?
3. Calculate the distance between (5, -3) and (-2, 1).
4. Points A and B are located at (10, 5) and (4, 13) respectively. What is the distance between A and B?
5. Determine the distance between (-7, -4) and (0, 2).
6. Find the distance between (1, 1) and (10, 10).
7. What is the distance between (-5, 8) and (3, -1)?

✅ Conclusion

The distance formula is a fundamental concept in coordinate geometry. Mastering it allows you to solve a wide range of problems, from finding distances between points to more complex geometric applications. Remember to practice and apply it in different scenarios to solidify your understanding!