Coordinates Meet Geometry

Lesson 8 of 10 · 7 min · Free, no account needed

What you will learn

The distance between two points is the hypotenuse of the right triangle between them: d = √(dx² + dy²).
The midpoint is the average: ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2).
With these two, a grid can check squares, find treasure and pick the nearest neighbour.

Your character is at (2, 3), the enemy at (8, 11). How far apart are they?
Bet before you measure.
Every slanted distance hides a right triangle.
Drag A and B. Make the distance exactly 5, then exactly 13, with the line slanted.
Where do they meet halfway?
Drag the flag by eye, then show the real midpoint. Two rounds.
Two formulas, both from pictures.
Distance formula: d = √((x₂ − x₁)² + (y₂ − y₁)²). It is Pythagoras with legs dx and dy.
Hero (2, 3) and enemy (8, 11): dx = 6, dy = 8, so d = √100 = 10.
Midpoint: average the x's and average the y's. They meet halfway at (5, 7).
Build a square that stands tilted.
Drag the corners onto grid points. All sides equal, both diagonals equal, and no side level or upright.
Four equal sides make a rhombus; equal diagonals as well make it a square. Distances on the grid check both.
Game: radar.
A treasure sits on a grid point. Each ping tells how far away it is. Find it in as few pings as you can.
An AI sorts a new critter by its nearest neighbour.
Tap the critter that is really nearest to the new one at (6, 6). Then Check.
A diagonal step of 2 and 2 is √8 ≈ 2.83, shorter than 3 straight across.
How far is (1, 2) from (7, 10)?
d = √(6² + 8²) = √100 = 10.
Speed round with your square.
Tap the midpoint of each side within 30 seconds.

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