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 non-collinear 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 axioms (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.

Collinear and Coplanar

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

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

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

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

$A$, $B$, and $C$ are coplanar:$A$, $B$, and $C$ are non-coplanar:

Here are more axioms to ponder:

    • A plane can be defined by three non-collinear points, or, using the definition of a line, by a line and a point not on the line. Through any non-collinear 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:

Problem: For any three non-collinear 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. 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, since they have the same size and shape.

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}$.

Example: Emmy is hanging a rectangular mirror in her bedroom and wants the left side to be $ 3$ feet from a wall and the right side to be $ 4$ feet from the other wall. 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: Since $ AF=AB+BD+DF=10$ and segments can be added/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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