2. Distance Between Two Points

The distance 𝑑 between two points 𝐴(𝑥₁,𝑦₁) and 𝐵(𝑥₂,𝑦₂) is given by:

𝑑 = √((𝑥₂ − 𝑥₁)² + (𝑦₂ − 𝑦₁)²)

The distance formula calculates the straight-line distance between two points on a Cartesian coordinate plane. The formula is derived from the Pythagorean theorem, which relates the lengths of the sides of a right triangle. In this case, if you imagine a right triangle formed by the two points 𝐴 and 𝐵, the difference in their x-coordinates (𝑥₂ − 𝑥₁) gives you the length of one side, and the difference in their y-coordinates (𝑦₂ − 𝑦₁) gives you the length of the other side. By squaring these lengths and adding them together, you can find the square of the hypotenuse (the distance between the points). Finally, taking the square root gives you the actual distance.

Think of finding a route between two locations on a map. The formula is like using a ruler to measure the shortest path between two points on that map, without having to account for any detours or turns you might take along a winding road.

📌 Example: Find the distance between 𝐴(2,3) and 𝐵(6,7):

𝑑 = √((6−2)² + (7−3)²) = √(16 + 16) = √32 = 4√2

In this example, we calculate the distance between points 𝐴(2,3) and 𝐵(6,7). First, we substitute the coordinates into the distance formula. The x-coordinates yield (6 - 2)², which equals 4² = 16, and the y-coordinates yield (7 - 3)², which also equals 4² = 16. Thus, we add these results together, giving us √(16 + 16) = √32. Since √32 can be simplified to 4√2, we conclude that the distance is approximately 5.66 units.

Imagine measuring the distance between two points in a park. If point 𝐴 is at coordinates (2,3) and point 𝐵 is at (6,7), you're basically determining how far apart two picnic spots are, as if you were walking the straight line connecting both spots.

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