Length and midpoint

Maths · Coordinate geometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A line segment is the part of a line between two points. Two things are asked about it: how long it is and where its middle is.

For the length, think of the segment as the longest side of a right-angled triangle. The other two sides are the difference in the x-coordinates and the difference in the y-coordinates. Pythagoras' theorem then gives the length.

The midpoint is halfway along, so its coordinates are the averages of the two x-coordinates and of the two y-coordinates. If you know the midpoint and one end, go the same step again to reach the other end.

Rules to remember

Length of the segment from (x₁, y₁) to (x₂, y₂) = √[(x₂ − x₁)² + (y₂ − y₁)²]
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
For the midpoint you add the coordinates; for the length you subtract them
If M is the midpoint of AB, then B = (2 × x of M − x of A, 2 × y of M − y of A)
Useful whole-number triangles: 3, 4, 5; 5, 12, 13; 6, 8, 10; 8, 15, 17

Common mistake

Subtracting for the midpoint or adding for the length. Also, a negative difference becomes positive when squared: (−15)² = 225, not −225.

Worked example

A is (−1, 4) and B is (7, −11). Find the midpoint and the length of AB.

Midpoint: ((−1 + 7)/2, (4 + (−11))/2) = (6/2, −7/2) = (3, −3.5).
Differences: 7 − (−1) = 8 and −11 − 4 = −15.
Length = √(8² + (−15)²) = √(64 + 225) = √289 = 17.

Answer: Midpoint (3, −3.5), length 17 units

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