Points in the coordinate system
Two points in a coordinate system: how far apart are they, where is the middle, and which line goes through both?
The explanation is shown. Quick: distance, midpoint and the line. Advanced: also angle of inclination, axis intercepts and the points along the segment.
Your numbers
Answer
Distance between the points
10
Midpoint
(1 ; 4)
The line through the points
y = 0.75x + 3.25
How to work it out
- 1Δx = 5 − -3 = 8 · Δy = 7 − 1 = 6
- 2|AB| = √(Δx² + Δy²) = √(64 + 36) = 10
- 3M = ((-3 + 5) / 2 ; (1 + 7) / 2) = (1 ; 4)
- 4a = Δy / Δx = 6 / 8 = 0.75 · b = y₁ − a·x₁ = 3.25
The formula
|AB| = √((x₂ − x₁)² + (y₂ − y₁)²)
The distance formula is Pythagoras: Δx and Δy are the legs.
M = ((x₁ + x₂)/2 ; (y₁ + y₂)/2)
The midpoint is the average of the coordinates.
In depth: more numbers and “what if”
More quantities
| Size | Value |
|---|---|
| Δx | 8 |
| Δy | 6 |
| Slope a | 0.75 |
| Angle of inclination | 36.9° |
| Crosses the y-axis at | (0 ; 3.25) |
| Crosses the x-axis at | (-4.33 ; 0) |
| Distance from A to (0 ; 0) | 3.16 |
| Distance from B to (0 ; 0) | 8.6 |
Points on the segment from A to B
| Share of the way | Point |
|---|---|
| 0 % | (-3 ; 1) |
| 25 % | (-1 ; 2.5) |
| 50 % | (1 ; 4) |
| 75 % | (3 ; 5.5) |
| 100 % | (5 ; 7) |
More: Geometry
- PythagorasFind the missing side, set out a right angle, check whether a room is square, and find the space diagonal.
- Area and perimeterRectangle, triangle, trapezium, circle and composite shapes — area and perimeter.
- Volume and surface areaBox, cylinder, cone, sphere, pyramid, prism, pipe and frustum — also in litres.
- The circleRadius, circumference and area from one measurement — plus sector, annulus, chord and rise.
- Right triangleSine, cosine and tangent: all sides and angles from two known values.
- AnglesThe third angle in a triangle and the angle sum of a polygon.
- Similar figuresScale factor and missing sides — area is multiplied by k², volume by k³.
- Any triangleThe sine and cosine rules: find the rest from SSS, SAS, ASA or SSA.
- TransformationsReflect, rotate and translate a figure on a coordinate grid — before and after.