Midpoint of a segment
The midpoint of a line segment is the point exactly halfway between its two endpoints. You find it by averaging the x-coordinates and averaging the y-coordinates.
Key idea
Midpoint = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
Length = √[ (x₂ - x₁)2 + (y₂ - y₁)2 ]
Length using Pythagoras
The length (distance) between two points comes from Pythagoras' theorem applied to the horizontal and vertical differences. Square each difference, add them, then take the square root.
Worked example
Find the midpoint and length of the segment joining (0, 0) and (3, 4).
Midpoint = ((0 + 3)/2, (0 + 4)/2) = (1.5, 2).
Length = √[(3 - 0)2 + (4 - 0)2] = √(9 + 16) = √25 = 5.
Working carefully with signs
Take care with negative coordinates: subtracting a negative becomes an addition. The length is always positive because differences are squared before adding.
Remember
- Midpoint uses averages; length uses squared differences.
- The order of the two points does not affect the length.
- A (3, 4, 5) or (5, 12, 13) pattern often gives a whole-number length.