Midpoints

The formula

• The midpoint of the segment joining $(x_1, y_1)$ and $(x_2, y_2)$ is:

$$M = \left(\frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2}\right)$$

• In words: average the $x$-coordinates, and average the $y$-coordinates.
• The midpoint is the point exactly halfway along the segment.

Why it is an average

• Halfway between 4 and 10 is 7 — the average of the two numbers.
• The same reasoning applies independently to the horizontal and vertical directions, which is why the formula treats $x$ and $y$ separately.
• This makes it easy to remember and hard to get wrong, provided you average rather than subtract.

Working through an example

• The midpoint of $A(3, 7)$ and $B(11, 1)$:

  • $x$: $\dfrac{3 + 11}{2} = \dfrac{14}{2} = 7$
  • $y$: $\dfrac{7 + 1}{2} = \dfrac{8}{2} = 4$
  • $$M = (7, 4)$$

• With negative coordinates, for $C(-5, 2)$ and $D(3, -8)$:

  • $x$: $\dfrac{-5 + 3}{2} = \dfrac{-2}{2} = -1$
  • $y$: $\dfrac{2 + (-8)}{2} = \dfrac{-6}{2} = -3$
  • $$M = (-1, -3)$$

• The midpoint can have non-integer coordinates. The midpoint of $(1, 2)$ and $(4, 5)$ is $(2.5, 3.5)$, and that is a perfectly good answer.

Working backwards to find an endpoint

• If you know one endpoint and the midpoint, you can find the other endpoint.

• The most reliable method is to use the formula and solve:

  • If $A(2, 3)$ and $M(7, 8)$, find $B(x, y)$.
  • $$\dfrac{2 + x}{2} = 7 \;\Rightarrow\; 2 + x = 14 \;\Rightarrow\; x = 12$$
  • $$\dfrac{3 + y}{2} = 8 \;\Rightarrow\; 3 + y = 16 \;\Rightarrow\; y = 13$$
  • $$B = (12, 13)$$

• A quicker route: the step from $A$ to $M$ is repeated to get from $M$ to $B$. Here the step is $(+5, +5)$, so $B = (7+5, 8+5) = (12, 13)$ ✓

• Multiply, do not just add. A common error is to give the midpoint's coordinates plus the first point's, or to halve rather than double.

What midpoints are used for

• Diagonals of a quadrilateral. If the diagonals of a quadrilateral share the same midpoint, they bisect each other — which proves the shape is a parallelogram.

• Perpendicular bisectors. The perpendicular bisector of a segment passes through its midpoint, so finding the midpoint is always the first step.

• The centre of a circle given the endpoints of a diameter.

• Medians of a triangle — a median joins a vertex to the midpoint of the opposite side.

• Note how often the midpoint is a stepping stone rather than the final answer. Recognising when a question needs one is much of the skill.

Proving a parallelogram

• The standard method:
  • Find the midpoint of one diagonal.
  • Find the midpoint of the other diagonal.
  • If they are the same point, the diagonals bisect each other, so the quadrilateral is a parallelogram.
• Take the diagonals, not the sides. In quadrilateral $PQRS$ the diagonals are $PR$ and $QS$ — the pairs of opposite vertices.