A route, a distance and a slope are different
How do coordinates determine distance without confusing it with direction?
Entering midway? A coordinate pair gives horizontal then vertical position. Subtract final minus initial coordinates to obtain changes.
Start with a concrete question
Move two points on a plane and compare a straight segment with a horizontal-then-vertical route.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
For A=(−2,−1) and B=(1,3), predict the straight distance and the axis-aligned path length.
Change one thing. Inspect what happens.
- Use the right-triangle legs to check the default distance.
- Swap A and B; compare distance, signed changes and midpoint.
- Make the segment vertical and explain why its distance exists while its slope is undefined.
- Make the points coincide and distinguish zero distance from a unique direction.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Segment and right-triangle decomposition — equal coordinate units
Horizontal: x coordinate (m). Vertical: y coordinate (m).
- 1. Straight segment A to B (solid line)
- 2. Horizontal then vertical path (dashed line)
- 3. Midpoint (points)
Axes use the displayed ranges.
Read plotted values as a table
| Series | x coordinate (m) | y coordinate (m) |
|---|---|---|
| Straight segment A to B | -2 | -1 |
| Straight segment A to B | 1 | 3 |
| Horizontal then vertical path | -2 | -1 |
| Horizontal then vertical path | 1 | -1 |
| Horizontal then vertical path | 1 | 3 |
| Midpoint | -0.5 | 1 |
Follow the calculation
- A=(-2, -1), B=(1, 3). Differences are B−A: (3, 4).
- Distance² = (3)² + (4)² = 25; distance = 5.00000 m.
- Midpoint = ((-2+1)/2, (-1+3)/2) = (-0.5, 1).
- Slope = vertical change / horizontal change = 4/3 = 1.33333.
Distance uses squared changes and cannot be negative. A signed slope is a direction ratio, not the length of the segment.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Start with the meaning
A coordinate pair gives horizontal then vertical position. Subtract final minus initial coordinates to obtain changes.
Move two points on a plane and compare a straight segment with a horizontal-then-vertical route.
A worked explanation
The Pythagorean theorem connects the squared legs of a right triangle to the squared hypotenuse. Coordinate changes supply the legs.
The default differences are 3 and 4, so straight distance is √(9+16)=5 m. An axis-aligned route is 3+4=7 m; the midpoint is (−0.5,1).
A slope divides vertical change by horizontal change. A vertical segment has zero denominator; inventing an infinite numeric slope would conceal that domain issue.
Symbols and relationship
d=√((Bx−Ax)²+(By−Ay)²); midpoint=((Ax+Bx)/2,(Ay+By)/2). Slope=Δy/Δx only when Δx≠0.
Use the graph and trace together
The graph and calculations use your current inputs. Read axis labels and units before comparing positions or slopes. Change one input, predict the effect, and use the numerical trace to verify the plotted behavior. A displayed curve is an illustration; the formula and its assumptions determine which conclusions are justified.
Scope and edge cases
- Flat Euclidean plane; physical barriers and curved surfaces are absent.
- Coordinates have equal units; slopes across unrelated axis units require their units to be retained.
Try a fresh case
What is the distance between (−1,2) and (5,10), in coordinate units?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Plan two routes on graph paper and compare straight and axis-aligned distances.
College investigation
Generalize distance to three dimensions and examine alternative metrics.
Workplace investigation
Document coordinate units and route constraints before using a straight-distance estimate.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.