Coordinate geometry
Once every point of the plane has a pair of numbers, can lengths, areas and even proofs be done by arithmetic instead of by eye?
Where you are 9 inside
What this area is about
Put a figure on a grid and each corner becomes a pair of numbers (x, y). Lengths, areas, midpoints and right angles can then be computed, with no ruler and no guessing from the drawing.
How these fit together
The setting. The coordinate plane gives every point its pair (x, y).
Lengths. Horizontal and vertical distance measures a segment that runs along a grid line by subtracting coordinates. The distance formula measures a slanted segment as the hypotenuse of a right triangle.
Points on a segment. The midpoint formula finds the point halfway between two points, and runs backwards to a missing endpoint. Partitioning a segment finds the point that cuts it in a ratio a : b.
Measuring figures. Area of a plotted polygon uses only grid-line lengths: a base and height read off the grid, or a box with its corner pieces taken away. Perimeter and area adds the distance formula, so slanted sides can be measured too.
Proving. Classifying a plotted quadrilateral tests slopes, lengths and midpoints to name the shape for certain. Placing a figure uses letters for coordinates, so one computation proves a theorem for every such figure.
A path through it
The coordinate plane, then horizontal and vertical distance and the area of a plotted polygon. Next the distance formula, then perimeter and area with slanted sides. Then the midpoint formula, partitioning, classifying quadrilaterals, and last placing a figure for a proof.
This area leads to the equation of a circle, coordinates in space and choosing a proof format.