Points, Lines, Planes, Line Segments, and Distance

Points, Lines, and Planes

Here are some basic definitions:

Point. A specific location in space, usually represented as a dot and capital letter. A point has zero dimensions.

Line. A collection of points that extend indefinitely in a straight formation. It can be represented by two points. Here is line  l or line $ \overleftrightarrow{{AB}}$ or $ \overleftrightarrow{{BA}}$ (order of points doesn’t matter):

Plane. A flat surface in space that extends indefinitely. It can be either be identified by a letter or three noncollinear points on the plane. Here is plane $ P$ or plane $ ABC$ (or any combination of the three letters):

Space: Boundless, three-dimensional set of all points (containing lines and planes).

See if these postulates (self-accepted statement) make sense. Visualize when thinking about them; geometry is all about visualization!

    • Any given point has an infinite number of lines that can be drawn through it.
    • Only one line can be drawn through two distinct points.
    • If two lines intersect, they do so at only one point.

We’ll learn more about these postulates, as well as other ones, in the Introduction to Proofs section.

We’ll learn later that two lines that don’t intersect are parallel, which means they are always the same distance apart, like railroad tracks.

Collinear and Coplanar

Three or more points are collinear if a line contains them. Otherwise, they are noncollinear.

$A$, $B$, and $C$ are collinear:               

$A$, $B$, and $C$ are noncollinear:     

Similarly, points and lines that lie in the same plane are coplanar. Otherwise, they are noncoplanar.

$A$, $B$, and $C$ are coplanar:$A$, $B$, and $C$ are noncoplanar:

Here are more axioms to ponder:

    • A plane can be defined by three noncollinear points, or, using the definition of a line, by a line and a point not on the line. Through any noncollinear points is there one and only one plane.
    • Exactly one plane passes through two intersecting lines.
    • The intersection of two planes is exactly one line.
    • If a line doesn’t lie in a plane but intersects it, that intersection is a point.

Here are some example Problems and Solutions:

Problem: For any three noncollinear points $ A$, $ B$, and $ C$, how many different lines can be drawn through different pairs of points?

Solution: In geometry, you’ll be drawing a lot to get your answers! Here’s what we can draw:

We can see that there are three lines we can draw: $ \overleftrightarrow{{AB}}$, $ \overleftrightarrow{{BC}}$, and $ \overleftrightarrow{{AC}}$.

Problem: Given this figure,

    1. How many planes are in the figure?
    2. Are line $ \overleftrightarrow{{AC}}$ and point $ F$ coplanar ? Why ?
    3. Are points $ C$, $ E$, and $ D$ coplanar ? Why ?
    4. Name three planes passing through at $ C$.

Solution :

    1. There are six planes, corresponding to the six sides of the figure
    2. Yes, because any line and a point not on the line are coplanar.
    3. Yes. Points $ C$ and $ E$ form a line and point $ D$ is not on that line. A line and any point not on the line are coplanar.
    4. One way to describe the three planes is $ ABC$, $ CDG$, and $ ACG$.

Line Segments and Distance

Line segments are part of a line and have fixed lengths with endpoints, which are typically used to name the segment. Here is line $ \overleftrightarrow{{AB}}$ ( or $ \overleftrightarrow{{BA}}$), followed by line segment $ \overline{{AB}}$ ( or $ \overline{{BA}}$). Note that lines have an infinite number of line segments:

The measure of segment $ \overline{{AB}}$ can be written just as $ AB$. The segments can be added if they share an endpoint. For example, $ AB+BC=AC$ (and $ AC-BC=AB$ and $ AC-AB=BC$):

Measures are real numbers, so all arithmetic operations can be used with them! For example, if $ AB=2$ and $ BC=2$, $ AC=4$. Makes sense! We can also say that the distance between$ A$ and $ B$ is $ 4$. Note also in our example, segments $ \overline{{AB}}$ and $ \overline{{BC}}$ are congruent (notation is $ \overline{{AB}} \displaystyle \cong \overline{{BC}}$ ), since they have the same size and shape. This is from the Definition of Congruence: if two line segments have the same length, they are congruent.

Here are the formulas for distance between and midpoint of two points, either on a number line, or in a coordinate plane:

Number Line:

The Distance between two points is the absolute value (non-negative value) of the difference of each of their coordinate points: $ AB=\left| {{{x}_{1}}-{{x}_{2}}} \right|\,\,\text{or}\,\,\left| {{{x}_{2}}-{{x}_{1}}} \right|$.

The Midpoint of two points is the exact middle point between them: $ \displaystyle \frac{{{{x}_{1}}+{{x}_{2}}}}{2}$.

Problem: Emmy is hanging a large rectangular mirror in her bedroom and she notices that she can have the left side of the mirror to be $ 3$ feet from one wall corner and the right side to be $ 4$ feet from the other wall corner, as shown in the figure below. She knows the total length of the wall is $ 10$ feet, and she’s using one nail to hang it. How wide is the mirror? Where should the nail be placed?

Solution: Notice that $ BD$ can represent the width of the mirror. Since $ AF=AB+BD+DF=10$, and segments can be added and subtracted if they share endpoints, $ BD=AF-(AB+DF)=10-7=3$ feet. This is the width of the mirror. To get the midpoint (where the nail should be placed), take the midpoint of $ BD$ (point $ C$), which equals $ \displaystyle \frac{1}{2}$ the distance between points $ B$ and $ D$, which is $ \displaystyle 1\frac{1}{2}$ feet. Thus, the nail should be placed $ \displaystyle 1\frac{1}{2}$ from either edge of the mirror. Makes sense!

Coordinate plane:

Distance and Midpoint formulas for the coordinate plane can be found here in the Coordinate Systems and Graphing Lines section.

On to Rays and Angles.

 

 

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