303. The Mid-Point Promise · always exactly halfway between

Try to break it

Drag A and B anywhere on the canvas. P always lands at ((x₁ + x₂)/2, (y₁ + y₂)/2) — the average of the coordinates. AP = PB at every position. Try to stretch AB so P drifts away from the centre; impossible.

How you build it

Mark the midpoint of segment AB.

• Place point A anywhere on the canvas.
• Place point B anywhere on the canvas.
• Draw the line segment connecting A and B.
• Construct the perpendicular bisector of AB to locate the midpoint P.

The proof, step by step

Prove that the midpoint of AB is ((x₁+x₂)/2, (y₁+y₂)/2).

• Let P(x, y) be the midpoint of the line segment joining A(x₁, y₁) and B(x₂, y₂).
• By definition, the midpoint divides the segment in the ratio 1 : 1.
• Using the section formula for the x-coordinate: x = (1·x₂ + 1·x₁) / (1 + 1) = (x₁ + x₂) / 2.
• Similarly, for the y-coordinate: y = (1·y₂ + 1·y₁) / (1 + 1) = (y₁ + y₂) / 2.
• Thus, the coordinates of the midpoint P are ((x₁ + x₂)/2, (y₁ + y₂)/2).

Worked example

The coordinates of the mid-point of the line segment joining the points (3, 4) and (7, 8) are:

Using the mid-point formula: x = (3+7)/2 = 5, y = (4+8)/2 = 6. Hence, the mid-point is (5, 6).

• (5, 6) — correct
• (4, 5)
• (10, 12)
• (2, 3)