Defining and Constructing Objects

Most of geometry is about working with geometric objects. Thus, it is necessary to have consensus on how the objects are defined. Here, some of the most fundamental geometric objects are defined.

Concept

Point

A point is a geometric object that cannot be constructed using simpler objects. Meaning, it is an undefined term, as opposed to a defined term. Instead, it has to be described using its characteristics. One possible description is that a point is an object with no size, indicating a location in space. It is represented graphically with a dot.

Concept

Line

A line, just like a point, is an undefined term. It is a one-dimensional object of infinite length with no width or height that never bends or turns. Its graphical representation is a straight line with arrowheads on either end, indicating that it continues indefinitely in both directions.

For each pair of non-identical points, it is possible to draw exactly one unique line through them. Thus, a line can be named using any two of the infinitely many points that lie on it. The above line could be named

\overleftrightarrow{AB},

\overleftrightarrow{BA},

or line

\ell.

A set of two or more points that lie on the same line is said to be collinear.

Concept

Line Segment

The line segment $\overline{AB},$ or segment $\overline{AB}$ , consists of the endpoints $A$ and $B,$ and every point on line $\overleftrightarrow{AB}$ located between $A$ and $B.$ Notice that, a segment, unlike a line, does not extend infinitely — it is bound by its endpoints.

Between two non-identical points, it's possible to draw exactly one unique segment.

Construction

Copying a Segment

Construction

Copying a Segment

Using a straightedge and a compass, it is possible to construct a copy of a segment. To do this, start by drawing a segment longer than the original with the straightedge.

To make the segments the same length, measure the original with a compass. Place the needle point on one endpoint, then, place the pencil at the other end.

The compass is now set to the correct length. After moving the compass to the new segment, mark where the end should be with the pencil.

This mark shows where to place the endpoint of the new segment. Placing the endpoint is the final step to copying the segment.

Concept

Plane

A plane is similar in its definition to a line, though it is two-dimensional. That is, it has an infinite width and length, but no height. It is represented by what could be seen as a floor or a wall, extending without end.

Through any three points not on the same line, exactly one plane can be drawn. This plane can be named using these points, as plane

ABC,

or alternatively as plane

P.

A set of points that are contained by the same plane are said to be coplanar.

Concept

Ray

A ray is a portion of a line, starting at an endpoint and extending indefinitely. The notation for a ray starts with the endpoint and another point on the ray, with an arrow drawn above it. For example, the ray below would be represented by: $\overrightarrow{AB}.$

The ray

\overrightarrow{AB}

lies on the line

\overleftrightarrow{AB},

starting at

A,

continuing through

B,

and then continuing indefinitely in that direction. Note that it's important to distinguish between the ray

\overrightarrow{AB}

and

\overrightarrow{BA};

although they both lie on the same line, they point in opposite directions and have different endpoints.

Name the objects

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Exercise

Give another name for ray $\overrightarrow{NP},$ line $l,$ and plane $M.$

Show Solution

Solution

The ray $\overrightarrow{NP}$ is the ray with endpoint $N$ that passes through $P,$ continuing infinitely.

This ray can also be named $r,$ as is indicated by the diagram. Continuing with line $l,$ we can see that it passes through the points $O$ and $Q.$

Therefore, we can name the line $\overleftrightarrow{OQ}.$ As $M$ is a plane, it can be named using three points in the plane that do not lie on the same line. There are three such points in this case, $N,$ $O,$ and $P.$

Thus, the plane can be named $NOP.$

Sketch the diagram

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Exercise

Sketch a diagram where plane $A$ and plane $B$ intersect along $\overleftrightarrow{PQ}.$

Show Solution

Solution

Let's start by sketching the plane $A,$ in any orientation we prefer.

Next, we can sketch the plane $B,$ so that $A$ and $B$ intersect along a line. By letting $B$ have some other orientation than $A,$ we'll accomplish this. Instead of having it lying down like a floor, we can draw $B$ as a wall.

These two planes now intersect along a line. For this line to have the name $\overleftrightarrow{PQ}$ as desired, the points $P$ and $Q$ must lie on the line. Let's draw these.

We have now finished the sketch.

Defining and Constructing Objects

Point

Line

Line Segment

Construction

Copying a Segment

Copying a Segment

Plane

Ray

Example

Name the objects

Example

Sketch the diagram