Coordinate Geometry • Topic 2 of 4

Midpoint Formula

The midpoint of a segment joining (x₁, y₁) and (x₂, y₂) is the average of the coordinates: ((x₁ + x₂)/2, (y₁ + y₂)/2). It is the point exactly halfway between the two endpoints. Add the x-coordinates and halve, then add the y-coordinates and halve. The formula also works in reverse: if you know the midpoint and one endpoint, you can solve for the other endpoint by doubling the midpoint coordinate and subtracting the known one. Watch the signs carefully when coordinates are negative. The SAT uses midpoints directly and inside problems about diagonals, bisectors and centres of figures.

Midpoint of (2,2) and (8,6) is (5,4)Midpoint formulaO(2, 2)(8, 6)(5, 4)Midpoint = ((2+8)/2, (2+6)/2) = (5, 4)

✅ Solved examples

1. Find the midpoint of (2, 4) and (6, 8).
((2+6)/2, (4+8)/2) = (4, 6).
2. Find the midpoint of (0, 0) and (10, 4).
(5, 2).
3. Find the midpoint of (−2, 3) and (4, 7).
((−2+4)/2, (3+7)/2) = (1, 5).
4. Midpoint of (1, 1) and (x, 5) is (3, 3). Find x.
(1 + x)/2 = 3, so x = 5.

✏️ Practice — try these, take hints as needed

1. Find the midpoint of (2, 2) and (8, 10).
Average each coordinate.
((2+8)/2, (2+10)/2).
—
(5, 6).
2. Find the midpoint of (0, 0) and (6, 12).
(3, 6).
—
—
(3, 6).
3. Find the midpoint of (−4, 2) and (2, 8).
((−4+2)/2, (2+8)/2).
—
—
(−1, 5).
4. Find the midpoint of (3, −1) and (7, 5).
((3+7)/2, (−1+5)/2).
—
—
(5, 2).
5. Midpoint of (2, 4) and (x, 4) is (5, 4). Find x.
(2 + x)/2 = 5.
2 + x = 10.
—
8.

📝 Topic test — 8 questions

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