Length and Midpoint of a Segment
The length of a segment is the Pythagorean distance between its endpoints, and the midpoint is the average of the x-coordinates and the average of the y-coordinates.
A complete interactive classroom, not just a preview.
Start when you are ready to enter this Stage's 4 scenes and explore, respond, and learn as you go.
How do you compute the length and midpoint of the segment from A(6, −1) to B(−3, 7)?
Two points A(6, −1) and B(−3, 7) sit on a coordinate plane. How long is the segment between them, and where exactly is its center?
Distance and midpoint look like different ideas, but both come from the same simple operation on coordinates. Picking the wrong formula gives an answer that is off by a fixed amount, and it is not obvious which step slipped.
Step-by-step computation of the distance using the Pythagorean theorem, then the midpoint by averaging coordinates, shown on a labeled coordinate sketch.
The length of AB is √(9² + 8²) = √145, and the midpoint is ((6 + (−3))/2, (−1 + 7)/2) = (1.5, 3), so distance is total horizontal and vertical change combined, and the midpoint is the average of each coordinate.
- Vector dot products
- Parametric line equations
- 3D coordinates
- 01A Segment Across Two QuadrantsslideSlot 1Hook
Plot A(6, −1) and B(−3, 7) on a coordinate plane and connect them. Ask how long the segment is and where its center lies.
- Point A is in Quadrant IV, point B is in Quadrant II
- The segment crosses both axes
- Two numbers are wanted: a length and a center point
PhenomenonTwo labeled points connected by a straight line that passes through the origin region.
QuestionHow do you find the length of AB and the point exactly halfway between A and B?
- 02Two Formulas, One Pair of PointsslideSlot 2Tension
Surface the common guess that length is just the difference of x-values, and that midpoint is the average of the points written as one number. Show why each is wrong.
- Subtracting x-values alone ignores the vertical change
- Writing the midpoint as a single number loses the y-coordinate
- Distance and midpoint need both coordinates, but in different ways
PredictionLength = |6 − (−3)| = 9 and midpoint = ((6 + (−3))/2) = 1.5.
Tempting intuitionTreating the segment as a one-dimensional line on the x-axis, because the x-difference is the larger number.
- 03Compute Length and MidpointinteractiveSlot 3Reveal
An interactive that shows the horizontal and vertical legs of the right triangle from A to B, then fills in the distance and midpoint formulas step by step.
- Horizontal leg: |6 − (−3)| = 9
- Vertical leg: |−1 − 7| = 8
- Distance: √(9² + 8²) = √145
- Midpoint: ((6 + (−3))/2, (−1 + 7)/2) = (1.5, 3)
EvidenceA right triangle with legs 9 and 8 dropped from A(6, −1) to B(−3, 7), and a plotted midpoint at (1.5, 3).
ConclusionLength of AB is √145 and the midpoint is (1.5, 3).
Mechanism- 1Form a right triangle by taking the horizontal change Δx = 6 − (−3) = 9 and vertical change Δy = −1 − 7 = −8, giving legs |9| and |8|.
- 2Apply the Pythagorean theorem: distance = √(Δx² + Δy²) = √(81 + 64) = √145.
- 3Average each coordinate independently: midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2) = ((6 + (−3))/2, (−1 + 7)/2) = (1.5, 3).
- 04Use the Same Two Numbers, Two WaysslideSlot 4Takeaway
Generalize: for any two points, build a right triangle from Δx and Δy to get the distance, and average each coordinate to get the midpoint.
- Distance uses square roots of squared differences
- Midpoint uses straight averages, no square roots
- Both formulas treat x and y separately
TransferGiven P(−2, 4) and Q(5, −3), find the distance and midpoint.
Expected inferenceΔx = 7, Δy = −7, so distance = √(49 + 49) = 7√2, and midpoint = (1.5, 0.5).
Discussion threads for a Stage aren't available yet.
