Orienting Yourself: The Use of Coordinates
Midpoints and trisection (enrichment)
Find the point halfway between two points, and the points that split a line into three.
The book's Problem 9 gives a table of points S, M and T and asks whether M is the midpoint of ST. The midpoint is the point exactly halfway along the segment.
"Halfway across and halfway up." Halfway in x is the average of the two x-coordinates; halfway in y is the average of the two y-coordinates. So the midpoint of (x1, y1) and (x2, y2) is ((x1 + x2)/2, (y1 + y2)/2).
It works backwards too. If you know the midpoint and one end, the other end is twice the midpoint minus the known end.
Trisection points cut a segment into three equal parts. From A to B, each third is one third of the move across and one third of the move up or down. A neat check: the two trisection points are two midpoints in disguise. P is the midpoint of A and Q, and Q is the midpoint of P and B.
These problems are marked * in the book: enrichment, not for examinations.
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