Point, Line and Plane
In short: Point, line and plane are the three undefined building blocks of geometry: a point marks a location, a line runs on forever in one dimension, and a plane is a flat surface running on forever.
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What is Point, Line and Plane?
Every figure in geometry rests on point, line and plane, three ideas accepted without a formal definition. A point has a position and no size. A line is a straight, thin path with no end in either direction. A plane is a perfectly flat surface with no edges and no thickness.
Two postulates tie them together. Two distinct points lie on exactly one line, and three points not on a common line lie on exactly one plane. Points on one line are collinear; points on one plane are coplanar.
How to spot it
- A problem names capital letters such as A, B and C and asks whether they sit on one straight path.
- A diagram shows a flat shaded parallelogram tagged with a single script letter, like plane M.
- A question asks how many lines pass through a set of points, or how many planes contain them.
- Words such as collinear, coplanar, intersect and contain appear in the statement.
The method
- Name each point with a capital letter and write the line through two of them as \(\overleftrightarrow{AB}\).
- Test collinearity: if a third point also lies on \(\overleftrightarrow{AB}\), the three are collinear; if not, they fix a plane.
- Count lines through \(n\) points, no three collinear, with \(\frac{n(n-1)}{2}\), since each pair gives exactly one line.
- Recall the intersections: two lines meet in at most one point, and two different planes meet in a line.
Point, Line and Plane: solved example
Problem
Six tent stakes are driven into a field in Oregon, and no three of them stand in a straight row. How many different straight lines can be drawn through pairs of stakes?
Solution
Two points determine exactly one line, so every pair of stakes gives one line. Count the pairs: \(\frac{6 \times 5}{2} = 15\). Dividing by 2 stops the pair A and B from being counted again as B and A.
Because no three stakes are collinear, no line is shared by two pairs. The field holds 15 lines.
Try it
Question 1
Points P, Q and R lie on one line, and point S is not on that line. How many planes contain all four points?
Show the answer
Exactly one. A line together with a point off it fixes a single plane, and P, Q, R and S all lie in it.
Question 2
Can four points in space fail to lie in a common plane?
Show the answer
Yes. Take three corners of a box’s floor and one corner of its ceiling: the floor corners fix a plane that misses the ceiling corner.
Point, Line and Plane vs. Line Segment
A line has arrows on both ends because it never stops, so it has no length. A line segment is only the piece between two endpoints and can be measured. In point, line and plane vocabulary, \(\overleftrightarrow{AB}\) is endless while \(\overline{AB}\) stops at A and B.
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Related words
Whole dictionary: A–Z · Geometry · Geometry