The Midpoint Formula

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Idea

The midpoint is the point exactly halfway along a segment. Finding it is simple: take the average of the two $x$ values, and the average of the two $y$ values.

Rule

For a segment with endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$ , the midpoint is

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

Averages, nothing more - and it works in every quadrant, for every slope.

How it is used

Three standard patterns:

Direct computation. Average the coordinates.
Recover an endpoint. If the midpoint is known, double it and subtract the known endpoint: $x_2 = 2x_0 - x_1$ . Each coordinate gives its own little equation.
Midpoint at the origin. Then the two endpoints sit opposite each other. Each pair of coordinates must add to zero. That gives two equations, and you never divide by $2$ . One more use: the median of a triangle goes from a corner to the midpoint of the opposite side. Find the midpoint first, then use the distance formula.

Worked example

Find the midpoint of $A(-4, 2)$ and $B(2, 8)$ . Average the abscissas: $\dfrac{-4 + 2}{2} = -1$ . Average the ordinates: $\dfrac{2 + 8}{2} = 5$ . So $M(-1, 5)$ - drag the points in the explorer below and watch M stay perfectly centered.

Explore

Drag either endpoint. The green point M is always the average of A and B. The tick marks show that the two halves stay equal. Now place A and B on opposite sides of the origin, at the same distance. M lands on the origin.

The midpoint

Try this. Move one endpoint two units: the midpoint moves one unit. Make the endpoints coincide to check the limiting case.

Point A · x-2Point A · y3Point B · x3Point B · y1

Midpoint M = ((-2 + 3)/2, (3 + 1)/2) = (0.5, 2)

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

What is the midpoint of

(-2,4)

and

(6,2)

$(2,3)$

$(4,6)$

$(4,-1)$

Hint 1 · Find a starting point

Average corresponding coordinates separately.

Hint 2 · Take the next step

The x average is (−2 + 6)/2 and the y average is (4 + 2)/2.

Show the reasoning

Answer: $(2,3)$

The midpoint is $(2,3)$ . Each coordinate is halfway between the endpoints; subtracting coordinates measures a displacement instead.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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