The middle of a segment

The idea

The mid-point of the join of A(x₁, y₁) and B(x₂, y₂) divides it in the ratio 1 : 1, so its coordinates are ((x₁ + x₂)/2, (y₁ + y₂)/2).

Meet in the middle

Two friends live at A(2, 4) and B(8, 10). They walk towards each other at the same speed along the straight path AB, and they meet at the middle of it.

How far is the meeting point from the y-axis (what is its x-coordinate)?

• 6
• 10
• 5

Move A and B, watch the middle

Drag A and B, or use the arrows. M is always the middle of AB. Watch what happens to M when only one end moves.

The mid-point formula

x = (x₁ + x₂)/2 y = (y₁ + y₂)/2 are the coordinates of the mid-point of the join of A(x₁, y₁) and B(x₂, y₂).

M cuts AB into two equal halves, so it divides AB in the ratio 1 : 1. Put m₁ = 1 and m₂ = 1 in the section formula: x = (1·x₁ + 1·x₂)/(1 + 1) = (x₁ + x₂)/2, and the same for y.

Each coordinate of M is the average of the same coordinates of A and B. That is why M moves only half as far as an end: move A right by 2 and M moves right by 1. For the friends, M = ((2 + 8)/2, (4 + 10)/2) = (5, 7).

Notes

x = (x₁ + x₂)/2 y = (y₁ + y₂)/2 is the mid-point of the join of A(x₁, y₁) and B(x₂, y₂): the section formula with m₁ = m₂ = 1.

Check yourself

Find the x-coordinate of the mid-point of A(−4, 2) and B(10, −6).

Answer: 3

x = (−4 + 10)/2 = 6/2 = 3.

Now find the y-coordinate of the same mid-point (A(−4, 2) and B(10, −6)).

Answer: -2

y = (2 + (−6))/2 = −4/2 = −2. So the mid-point is (3, −2).

A(6, 1), B(8, 2), C(9, 4) and D(p, 3) are the vertices of a parallelogram ABCD, taken in order. Its diagonals AC and BD bisect each other, so they have the same mid-point. Find p.

Answer: 7

The mid-point of AC is (15/2, 5/2) and of BD is ((8 + p)/2, 5/2). So (8 + p)/2 = 15/2, 8 + p = 15, and p = 7.

M(4, 6) is the mid-point of AB, and A is (1, 2). What are the coordinates of B?

• (2.5, 4). That is the mid-point of A and M. M is the mid-point of A and B, so B is as far beyond M as A is before it.
• (7, 10) — correct. Yes! x₁ + x₂ = 2 × 4 = 8 gives x₂ = 7, and y₁ + y₂ = 2 × 6 = 12 gives y₂ = 10. Check: ((1 + 7)/2, (2 + 10)/2) = (4, 6).
• (5, 8). That is A + M, (1 + 4, 2 + 6). Use x₁ + x₂ = 2 × 4 = 8 to find x₂, and y₁ + y₂ = 2 × 6 = 12 to find y₂.