Points, Lines, and Planes - Interactive Activity

Definitions and Main Ideas

Geometry begins with three undefined terms — point, line, and plane — that are described rather than defined. From them we build segments, rays, and angles, and we describe how points, lines, and planes relate: points on the same line are collinear, points in the same plane are coplanar, and lines and planes meet at intersections. This lesson sets the vocabulary every later lesson depends on.

Point

A specific location in space. A point has no dimension; it is represented by a dot and named with a capital letter.

Line

A series of points that follows a straight path and extends forever in both directions. A line has no thickness; it is named by a lowercase script letter or by any two points on it. The line through points A and B, for example, is named line AB, with the double-headed arrow over the letters.

Line Segment

A section of a line with two defined endpoints. A segment is named by its two endpoints: the segment with endpoints A and B is named segment AB, with the bar over the letters.

Ray

A part of a line that has one defined endpoint and extends forever in one direction. A ray is named by its endpoint followed by another point on the ray: the ray with endpoint A that passes through B is named ray AB, with the arrow over the letters pointing from the endpoint toward the second point.

Opposite Rays

Two rays that share an endpoint and extend in opposite directions, together forming a line.

Plane

A flat surface that extends forever in all directions and has no thickness. A plane is named by a capital script letter or by any three noncollinear points in it. The plane with script letter M, for example, is named plane M, with the small parallelogram symbol before the letter.

Collinear Points

Points that lie on the same line. Any two points are collinear; the term describes a third point, or a group of points, in relation to a line.

Coplanar

Points or lines that lie in the same plane. Points or lines that do not all lie in one plane are noncoplanar.

Main ideas

1. Points, lines, and planes are the building blocks of geometry. A point has no size, a line has length but no thickness, and a plane is a flat surface with no thickness.
2. Two points determine a line, and three noncollinear points determine a plane.
3. Two lines intersect at exactly one point. Two planes intersect at a line. A line can intersect a plane at a single point.
4. Segments have endpoints and can be measured; lines and rays extend forever and cannot be measured.
5. A point lies on a line given by an equation exactly when its coordinates satisfy the equation — substituting the coordinates is how collinearity with a graphed line is tested.

Figure Explorer

What you are looking at: Plane M, drawn as a flat surface, with line l inside it. Points A, B, and C lie on line l. Point D also lies in plane M but not on line l, and point E lies outside plane M. The gold highlight shows the figure named by the button you choose.

What to do: Choose a button under the diagram to highlight that kind of figure and read about it. You can also drag the orange handle on point D: the readout tells you, from the position of D, whether D is collinear with A and B and whether D is coplanar with A, B, and C. Drag D onto line l and it settles exactly on the line.

What to notice: A segment stops at its two endpoints, a ray stops at one endpoint and extends forever the other way, and a line extends forever in both directions. The arrows in the diagram show which figures keep going.

Intersections Explorer

What you are looking at: One of the three ways lines and planes can intersect. The gold highlight marks the intersection itself: a single point, or a line.

What to do: Choose a button under the diagram to switch the scene, and read in the readout what the intersection is and why.

What to notice: Two lines intersect at a point, a line and a plane intersect at a point, and two planes intersect at a line. An intersection is always smaller than the figures that form it: points and lines have no thickness, even when they come from planes.

Name It

What to do: Each problem shows one figure. Work in two steps: (1) choose the name that describes the figure, and (2) choose the naming statement that is correct for it. The diagram comes first, then the problem statement, then the step question. The gold highlight in the diagram marks the figure the current step asks about, and a part you have already settled stays labeled in the problem statement. The choice you pick is highlighted in the diagram whether it is right or wrong, so you can see what the choice means. A first miss gives a hint and another try, with the missed choice removed. A second miss on a step names the correct choice and explains it, and the activity then moves on. Press New Item for the next problem: the five problems take turns in order, and the problem number is shown with the step question.