Coordinate Geometry Basics: Midpoint of a Segment and Coordinate Averages

1) Visual meaning of a midpoint: “halfway” as equal distance from both ends

Given a segment with endpoints A and B, the midpoint is the point on the segment that splits it into two smaller segments of equal length. If the midpoint is M, then the defining idea is:

• AM = MB (same distance from each endpoint)
• M lies on the segment from A to B (not off to the side)

A helpful way to picture this is “balance”: if you imagine the segment as a rigid bar with equal weights at A and B, the balance point is exactly the midpoint. On a coordinate grid, that balance idea turns into “average the coordinates.”

Midpoint on a horizontal or vertical segment (quick intuition)

If A and B line up horizontally (same y), the midpoint has the same y and its x-value is halfway between the two x-values. If they line up vertically (same x), the midpoint has the same x and its y-value is halfway between the two y-values. The general midpoint formula does the same thing in both directions at once.

2) Deriving the midpoint formula as coordinate averages

Let the endpoints be A(x1, y1) and B(x2, y2). The midpoint M should be halfway in the horizontal direction and halfway in the vertical direction.

Step-by-step derivation

• Horizontal (x) direction: The total change in x from A to B is x2 − x1. Half of that is (x2 − x1)/2. Starting at x1 and moving half the way gives: xM = x1 + (x2 − x1)/2.
• Simplify: xM = (2x1 + x2 − x1)/2 = (x1 + x2)/2.
• Vertical (y) direction: Similarly, yM = y1 + (y2 − y1)/2 = (y1 + y2)/2.

So the midpoint is:

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M( (x1 + x2)/2 , (y1 + y2)/2 )

This is why the midpoint is often described as the average of the endpoints’ coordinates: average the x-values to get the midpoint’s x, and average the y-values to get the midpoint’s y.

Geometric check idea (without re-teaching distance)

Once you compute M, you can do a quick “halfway” check by comparing coordinate changes: from A to M the change in x should be half of the change from A to B, and the change in y should also be half. In symbols:

• xM − x1 = (x2 − x1)/2
• yM − y1 = (y2 − y1)/2

If both are true, M is exactly halfway in both directions, which places it halfway along the segment.

3) Practice: find midpoints, plot them, and confirm visually

In each example: (1) compute the midpoint using averages, (2) plot A, B, and M, (3) confirm that M is halfway by checking the coordinate changes from A to M versus A to B.

Example 1 (mixed coordinates)

Let A(2, 1) and B(8, 7).

1. Compute midpoint

   xM = (2 + 8)/2 = 10/2 = 5
yM = (1 + 7)/2 = 8/2 = 4
   So M(5, 4).

2. Visual/halfway check using changes

   From A to B: Δx = 8 − 2 = 6, Δy = 7 − 1 = 6.
   Half of each is 3 and 3.
   From A to M: 5 − 2 = 3, 4 − 1 = 3.
   Matches half in both directions, so M is halfway along the segment.

Example 2 (negative coordinates)

Let A(-4, 6) and B(2, -2).

1. Compute midpoint

   xM = (-4 + 2)/2 = (-2)/2 = -1
yM = (6 + (-2))/2 = 4/2 = 2
   So M(-1, 2).

2. Visual/halfway check using changes

   From A to B: Δx = 2 − (-4) = 6, Δy = -2 − 6 = -8.
   Half is 3 and -4.
   From A to M: -1 − (-4) = 3, 2 − 6 = -4.
   Matches half in both directions.

Example 3 (horizontal segment)

Let A(-3, 5) and B(7, 5).

1. Compute midpoint

   xM = (-3 + 7)/2 = 4/2 = 2
yM = (5 + 5)/2 = 10/2 = 5
   So M(2, 5).

2. Visual meaning

   The y-value stays 5 (same horizontal line). The midpoint’s x is the average of -3 and 7, so it lands halfway between them.

Quick practice set (compute M)

4) Extension: given one endpoint and the midpoint, solve for the other endpoint

The midpoint formula can be reversed to find an unknown endpoint. Suppose:

• Known endpoint A(x1, y1)
• Midpoint M(xM, yM)
• Unknown endpoint B(x2, y2)

Start from the midpoint equations:

xM = (x1 + x2)/2 and yM = (y1 + y2)/2

Solve each for the unknown coordinate:

• 2xM = x1 + x2 so x2 = 2xM − x1
• 2yM = y1 + y2 so y2 = 2yM − y1

Example 4 (find the missing endpoint)

Given A(1, 3) and midpoint M(4, -1), find B.

1. Use the reversed formulas

   x2 = 2xM − x1 = 2(4) − 1 = 8 − 1 = 7
y2 = 2yM − y1 = 2(-1) − 3 = -2 − 3 = -5
   So B(7, -5).

2. Quick check

   Average the endpoints: (1+7)/2 = 4 and (3+(-5))/2 = -1, which matches M.

Example 5 (unknown endpoint with negatives)

Given A(-6, 2) and midpoint M(-1, 6), find B.

x2 = 2(-1) - (-6) = -2 + 6 = 4
y2 = 2(6) - 2 = 12 - 2 = 10
B(4, 10)

Geometric check idea: midpoint as a “symmetry partner”

If M is the midpoint, then A and B are symmetric with respect to M. The vector (change) from M to A is the opposite of the vector from M to B. Algebraically, that’s exactly what x2 = 2xM − x1 and y2 = 2yM − y1 accomplish: they “reflect” A across M to reach B.