CHAPTER 1 · RESOURCES

HALFWAY — Concept Note

Think first. Use AI purposefully. Verify always.

A concise revision of the approved lesson. For discovery, interaction and guided reasoning, return to the lesson.

CONCEPT

Midpoint of a Line Segment

Average x-coordinates together and y-coordinates together.
The midpoint lies on the segment and is equally far from both endpoints.
Use the midpoint relationship and solve backwards: repeat the change from the known endpoint to the midpoint.

M (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})

Add corresponding coordinates and halve each sum. Plot the result. Verify collinearity and equal endpoint distances.

One interval = one unit. M is the midpoint of AB.

A(-2, -1), B $(2, 3)$ . Horizontal change 4; vertical change 4. AB ≈ 5.66 units. M $(0, 1)$ . AM = MB ≈ 2.83.

x: -2 → 0 → 2
y: -1 → 1 → 3
AM = MB. Half the change in each coordinate.

Find the midpoint of A(−2, −1) and B(2, 3).

The average y-coordinate is (−1 + 3) ÷ 2 = 1, so M(0, 1).

Average corresponding coordinates. Equal distances alone do not establish that a point lies on the segment.

A(−2, 1) and M(1, 3) are an endpoint and midpoint. Find B.

Check your reasoning

B(4, 5): 2(1) − (−2) = 4 and 2(3) − 1 = 5.