Midpoint and Distance Between Two Points

The midpoint of a line segment is the point exactly halfway between its endpoints. The distance formula uses Pythagoras' theorem to find the straight-line distance between any two points on a coordinate grid.

The Midpoint Formula

The midpoint $M$ of a segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is:

Midpoint Formula

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

Average the x-coordinates and average the y-coordinates

Think of it as: add the two x-values and divide by 2; add the two y-values and divide by 2.

Find the midpoint of A(2, 4) and B(8, 10)

$M = \left(\frac{2+8}{2},\; \frac{4+10}{2}\right) = (5, 7)$

The Distance Formula

The distance $d$ between $(x_1, y_1)$ and $(x_2, y_2)$ comes from Pythagoras' theorem:

Distance Formula

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Square the difference in x, add the square of the difference in y, then take the square root

Find the distance between A(1, 2) and B(4, 6)

Difference in x: $4 - 1 = 3$

Difference in y: $6 - 2 = 4$

$d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Note

The order of subtraction doesn't matter because you square the differences. $(x_2 - x_1)^2 = (x_1 - x_2)^2$ .

Connecting the Two Formulas

The midpoint is another shape: the midpoint of $A(1,2)$ and $B(4,6)$ is $(2.5, 4)$ , and the distance from that midpoint to either endpoint is $5/2 = 2.5$ units.

Real world: planning a rest stop

Two towns are at coordinates A(10, 20) and B(50, 80) on a map grid (km). The midpoint is $M = \left(\frac{10+50}{2},\; \frac{20+80}{2}\right) = (30, 50)$ .

Checking the map at (30, 50) helps plan a refuelling stop roughly halfway.

Real world: checking a room diagonal

A builder measures a rectangular room 4 m × 3 m. The diagonal should be $d = \sqrt{4^2 + 3^2} = \sqrt{25} = 5$ m. If the measured diagonal differs, the room is not perfectly rectangular.

Real world: sports GPS tracking

A footballer's GPS shows positions at (12, 5) and (18, 13) in metres. The sprint distance is $d = \sqrt{6^2 + 8^2} = \sqrt{100} = 10$ m. Analysts sum hundreds of such segments to calculate total distance run.

Practice Problems

Find the midpoint of P(−3, 5) and Q(7, −1).

Show answer

M = ((−3+7)/2, (5+(−1))/2) = (4/2, 4/2) = (2, 2)

Find the distance between C(0, 0) and D(5, 12).

Show answer

d = √(5² + 12²) = √(25 + 144) = √169 = 13

M(4, 3) is the midpoint of AB. A is at (1, −1). Find B.

Show answer

4 = (1 + xB)/2 → xB = 7. 3 = (−1 + yB)/2 → yB = 7. So B = (7, 7).