Section 3

The Midpoint of a Line Segment

The midpoint of a line segment is the point that lies on the segment and is equidistant from its two endpoints. On the coordinate plane you do not need a ruler: the midpoint is simply the average of the two x-coordinates and the average of the two y-coordinates.

Definition

The Midpoint Formula

If A(x₁ , y₁) and B(x₂ , y₂), then the midpoint M of AB is M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

Average the x's

Add the two x-coordinates and divide by 2. This gives the horizontal position of M.

Average the y's

Add the two y-coordinates and divide by 2. This gives the vertical position of M.

Order does not matter

Addition is commutative, so starting from A or from B gives exactly the same midpoint.

xy-6-6-4-4-2-2224466A(-4,-2)B(2,4)M(-1,1)

x = (-4 + 2) / 2 = -2 / 2 = -1

y = (-2 + 4) / 2 = 2 / 2 = 1

So M(-1 , 1). Notice that M is exactly halfway along the drawn segment: the distance from A to M equals the distance from M to B.

Practice with me

Worked Examples

Example 1

Find the midpoint of the segment joining A(3 , 5) and B(7 , 1).

  1. 1Write the formula: M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 ).
  2. 2Substitute: M = ( (3 + 7)/2 , (5 + 1)/2 ).
  3. 3Simplify: M = ( 10/2 , 6/2 ).
  4. 4Answer: M(5 , 3).
xy-8-8-6-6-4-4-2-222446688A(3,5)B(7,1)M(5,3)
Example 2

M(2 , -1) is the midpoint of AB, and A(-3 , 4). Find the coordinates of B.

  1. 1The midpoint of the x's gives: (-3 + x₂)/2 = 2.
  2. 2Multiply both sides by 2: -3 + x₂ = 4, so x₂ = 7.
  3. 3The midpoint of the y's gives: (4 + y₂)/2 = -1.
  4. 4Multiply both sides by 2: 4 + y₂ = -2, so y₂ = -6.
  5. 5Answer: B(7 , -6).
xy-8-8-6-6-4-4-2-222446688A(-3,4)M(2,-1)B(7,-6)
Example 3

A segment has endpoints on the axes: A(0 , 6) and B(-8 , 0). Find its midpoint.

  1. 1x = (0 + (-8)) / 2 = -8/2 = -4.
  2. 2y = (6 + 0) / 2 = 6/2 = 3.
  3. 3Answer: M(-4 , 3) — it lies in the 2nd quadrant.
xy-8-8-6-6-4-4-2-222446688A(0,6)B(-8,0)M(-4,3)

Application

Midpoints and Quadrilaterals

In a parallelogram (and therefore in a rectangle, a rhombus and a square) the two diagonals bisect each other. That means the two diagonals share the same midpoint — a very useful test.

xy-6-6-4-4-2-2224466A(-4,1)B(0,4)C(4,1)D(0,-2)M(0,1)

Midpoint of AC = ( (-4 + 4)/2 , (1 + 1)/2 ) = (0 , 1)

Midpoint of BD = ( (0 + 0)/2 , (4 + (-2))/2 ) = (0 , 1)

The diagonals AC and BD have the same midpoint (0 , 1), so ABCD is a parallelogram. Here the diagonals are also perpendicular, so ABCD is in fact a rhombus.

Remember

Special Cases & Tips

  • A horizontal segment (same y for both endpoints) keeps that y: only the x is averaged.
  • A vertical segment (same x for both endpoints) keeps that x: only the y is averaged.
  • The midpoint may have a fraction: A(1 , 2) and B(4 , 3) give M(2.5 , 2.5). This is correct — the midpoint does not have to be a lattice point.
  • Always divide by 2 at the end; a common mistake is to report the sum instead of the average.
  • If the midpoint is the origin (0 , 0), the two endpoints are opposites: B = (-x₁ , -y₁).

Before you move on

Quick Check

  1. 1. Find the midpoint of A(-6 , 2) and B(2 , 8). Answer: (-2 , 5)
  2. 2. M(0 , 3) is the midpoint of CD with C(-5 , 1). Find D. Answer: (5 , 5)
  3. 3. Find the midpoint of the segment joining (7 , -4) and (7 , 10). Answer: (7 , 3)
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