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Hey there! Today we are going to explore how to find the exact distance between any two points on a flat grid, just like finding the shortest path on a map!
Imagine you are looking at a treasure map laid out on a grid. If you want to walk from one point to another, you could walk horizontally and then vertically, forming a right-angled triangle.
By connecting our two points with a straight line, we create the hypotenuse of a right-angled triangle. We can use the famous Pythagorean Theorem (a² + b² = c²) to help us find this diagonal distance!
The Distance Formula is just a neat package for this idea! It is written as: d = √((x₂ - x₁)² + (y₂ - y₁)²). We subtract the x-coordinates to find the width, subtract the y-coordinates to find the height, square them both, add them, and take the square root.
Find the distance between the points P(2, 3) and Q(6, 6).
- Identify the coordinates: (x₁, y₁) = (2, 3) and (x₂, y₂) = (6, 6).
- Find the horizontal distance: x₂ - x₁ = 6 - 2 = 4.
- Find the vertical distance: y₂ - y₁ = 6 - 3 = 3.
- Plug these values into the distance formula: d = √((4)² + (3)²).
- Simplify the squares: d = √(16 + 9) = √(25).
- Take the square root to get the final answer: d = 5 units.
