Orienting Yourself: The Use of Coordinates
Every point has an address of two numbers — and distances come from Pythagoras.
A point's coordinates (x, y) are instructions: across first, up second. The distance between two points is √((x₂−x₁)² + (y₂−y₁)²) — Pythagoras in disguise — and the midpoint is simply the average of the coordinates.
Every point on a map needs two answers — how far across, how far up. The Cartesian plane turns that idea into mathematics.
You'll learn to name any point with a pair of numbers, then use those pairs to measure distances and find midpoints with nothing but arithmetic.
This is the bridge between algebra and geometry: once shapes become coordinates, every geometry question can be attacked with calculation.
An address made of two numbers
A point's coordinates (x, y) are instructions: start at the origin O, go x steps horizontally, then y steps vertically. The order matters — (2, 5) and (5, 2) are different points.
The four quadrants
The two axes cut the plane into four quadrants. Signs of the coordinates tell you where you are: (+, +) is Quadrant I, (−, +) II, (−, −) III, (+, −) IV. A point on an axis has one coordinate 0.
Distance is Pythagoras in disguise
The horizontal gap |x₂ − x₁| and the vertical gap |y₂ − y₁| are the two legs of a right triangle, so d = √((x₂ − x₁)² + (y₂ − y₁)²). Horizontal or vertical distances are simply the difference of the matching coordinates.
The midpoint is the average
The point halfway between two others has coordinates that are the averages: ((x₁ + x₂)/2, (y₁ + y₂)/2). Averaging works because 'halfway' means 'equal shares of both gaps'.
These are easy on purpose — they check understanding, not speed. Nothing here is scored.
1. In which quadrant does (−2, −7) lie?
2. The midpoint of (2, 3) and (6, 7) is…
3. Distance between (1, 2) and (1, 6) is…
In which quadrant does (−2, −7) lie?