The midpoint of a segment is found by averaging the x-coordinates and averaging the y-coordinates. The same idea, run backwards, finds a missing endpoint, and with thirds instead of halves it trisects a segment.
Halfway between 4 and 10 on a number line is 7. The calculation is an average: (4 + 10) ÷ 2 = 7.
In the plane, the x- and y-directions are independent. So each direction is handled on its own. The midpoint’s x-coordinate is the average of the two x-coordinates. Its y-coordinate is the average of the two y-coordinates.
M = ( (x₁ + x₂) ÷ 2 , (y₁ + y₂) ÷ 2 )
Find the midpoint of S (−6, 2) and T (4, 6).
x: (−6 + 4) ÷ 2 = −2 ÷ 2 = −1
y: (2 + 6) ÷ 2 = 8 ÷ 2 = 4
M is (−1, 4). The negative coordinate needs no special rule. The average of −6 and 4 is −1, the number halfway between them on the line.
Two separate averages. The matching tick marks show that SM and MT are equal in length. Equal halves on the same straight line is what defines the midpoint.
The midpoint is defined as the point on the segment that is the same distance from both ends. The distance formula can check this for S (−6, 2), M (−1, 4) and T (4, 6):
The two distances are equal. Both moves are 5 right and 2 up, so M also lies on the straight line from S to T.
Sometimes the midpoint and one endpoint are given, and the other endpoint is asked for. The same rule is used with the unknown in it.
M (−7, 1) is the midpoint of A (3, −4) and B (x, y). For the x-coordinate:
(3 + x) ÷ 2 = −7 so 3 + x = −14 so x = −17
For the y-coordinate:
(−4 + y) ÷ 2 = 1 so −4 + y = 2 so y = 6
B is (−17, 6). Multiplying both sides by 2 first removes the fraction.
Check the answer another way. A is at x = 3 and M is at x = −7. Is x = −17 for B reasonable?
From A to M, x changes by −7 − 3 = −10. M is halfway, so from M to B, x must change by −10 again. −7 − 10 = −17. The answer agrees.
A second method that reaches the same answer is a check on the first. If the algebra had given −11, this check would show the error.
The trisection points of a segment AB are the two points P and Q that divide it into three equal parts. P is nearer A, Q is nearer B.
The midpoint can be described as: start at A and move half of the way to B. P is found by moving one third of the way:
P = ( x₁ + (x₂ − x₁)/3 , y₁ + (y₂ − y₁)/3 )
Q is found by moving two thirds of the way. For A (4, 7) and B (16, −2), the x-difference is 12 and the y-difference is −9. P is (4 + 4, 7 − 3) = (8, 4). Q is (4 + 8, 7 − 6) = (12, 1).
Any fraction of the way along a segment is found the same way: add that fraction of each coordinate difference to the starting point.
1The midpoint of (2, 6) and (8, 10) is ____.
2The midpoint of a segment is found by…
3In each case, is M the midpoint of S and T?
4M (5, 5) is the midpoint of A (2, 3) and B. Where is B?
5Why does averaging each coordinate give the point exactly halfway along the segment?
6The midpoint of a segment is (0, 0). What does that tell you about the endpoints?
7P and Q trisect the segment from A (4, 7) to B (16, −2), with P nearer A. Where is P?
Next: using coordinates to solve problems about rooms, maps and circles.
Noescio