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Hey there! Today we're going to learn how to find the exact distance between any two points on a grid using a clever math shortcut. Let's dive in!
Imagine you are looking at a map drawn on a grid. If you want to walk from one point to another, you could walk along the grid lines, or you could take a straight-line shortcut directly between them.
When we connect two points on a grid with a straight line, we can build a right-angled triangle underneath it. The horizontal distance is the change in our x-coordinates, and the vertical distance is the change in our y-coordinates.
By using the famous Pythagorean Theorem (a² + b² = c²), we can find the direct distance. In coordinate geometry, we write this as the Distance Formula: d = √((x₂ - x₁)² + (y₂ - y₁)²).
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).
- Calculate the horizontal change (difference in x): x₂ - x₁ = 6 - 2 = 4.
- Calculate the vertical change (difference in y): y₂ - y₁ = 6 - 3 = 3.
- Plug these values into the distance formula: d = √(4² + 3²).
- Simplify inside the square root: d = √(16 + 9) = √25.
- Take the square root to find the final distance: d = 5 units.
