The big picture
One right triangle gives the distance between two points. One formula gives a point at any ratio.
| Idea | Remember | Example |
|---|---|---|
| The distance formula | Square both gaps. Add them. Take the square root. | (1, 2) to (4, 6) gives 5 |
| Collinear, and k | Collinear when the two shorter distances add to the longest. For a missing k, set the squared distance equal to the given distance squared. | (2, k) is 5 from (−1, 3): k = 7 or −1 |
| Counting hops | Make m + n equal hops. Walk m of them from the first point. | 2 : 1 is 2 of 3 hops |
| The section formula | B's numbers carry m. A's numbers carry n. Both sit over m + n. | (9, 7) |
| Reading the ratio back | Compare the across gaps and the up gaps. Both must give the same ratio. Skip a pair of gaps that are both 0. | 2 : 1 |
| The midpoint | Average each coordinate. It is the ratio 1 : 1. | (2, 4) |
| The missing corner | D = A + C − B, because the diagonals share one midpoint. | (4, 5) |
| Cutting into thirds | Two points, at 1 : 2 and 2 : 1. | (5, 7) and (8, 11) |
| Equally far | Set the two squared distances equal. | (0, −3) |
| Name the shape | Find every distance. Compare them. The pattern names the shape. | sides 4, 3, 4, 3 and diagonals 5, 5: rectangle |
This is Chapter 7, Coordinate Geometry. Open it and try its exercises. They ask these same questions.