Coordinates and midpoints

Coordinates give a point's position as (x, y): how far across, then how far up. The point (0, 0), where the axes cross, is the origin. Negative x is left of the origin, and negative y is below it.

1. Start at the origin.
2. Move across by the x-coordinate (right if positive, left if negative).
3. Move up or down by the y-coordinate.

Example: three corners of a square are (1, 1), (1, 4) and (4, 4). Find the fourth.

1. The sides are 3 long: from x = 1 to x = 4, and from y = 1 to y = 4.
2. The missing corner is below (4, 4) and level with (1, 1).
3. (4, 1)

The midpoint of a line segment is halfway between its ends. Find the mean of the x-coordinates and the mean of the y-coordinates.

Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Example: find the midpoint of A(2, −3) and B(8, 5).

1. x: (2 + 8) ÷ 2 = 5
2. y: (−3 + 5) ÷ 2 = 1
3. (5, 1)

Harder example: M(3, 4) is the midpoint of PQ, and P is (−1, 6). Find Q.

1. From P to M, x goes up by 3 − (−1) = 4, and y goes down by 6 − 4 = 2.
2. Q is the same step on from M: x = 3 + 4 = 7, y = 4 − 2 = 2
3. Q(7, 2)

If AP : PB = m : n, then P is m/(m + n) of the way from A to B.

1. Find the changes in x and y from A to B, and multiply each by m/(m + n).
2. Add the results to A's coordinates.

Example: A is (1, 2) and B is (10, 14). Find P on AB with AP : PB = 1 : 2.

1. P is ⅓ of the way. Changes: 9 × ⅓ = 3 and 12 × ⅓ = 4
2. P = (1 + 3, 2 + 4) = (4, 6)

Harder example: A(−2, 5), B(8, 0) and AP : PB = 2 : 3. Find P.

1. P is 2/5 of the way. Changes: 10 × 2/5 = 4 and −5 × 2/5 = −2
2. P = (−2 + 4, 5 − 2) = (2, 3)