Defining and Constructing Objects

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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 AB,\overleftrightarrow{AB}, BA,\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 AB,\overline{AB}, or segment AB\overline{AB}, consists of the endpoints AA and B,B, and every point on line AB\overleftrightarrow{AB} located between AA and B.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.

Concept Plane.svg
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,ABC, or alternatively as plane P.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: AB.\overrightarrow{AB}.

The ray AB\overrightarrow{AB} lies on the line AB,\overleftrightarrow{AB}, starting at A,A, continuing through B,B, and then continuing indefinitely in that direction. Note that it's important to distinguish between the ray AB\overrightarrow{AB} and BA;\overrightarrow{BA}; although they both lie on the same line, they point in opposite directions and have different endpoints.
Exercise

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

Skills name the objects.svg
Solution

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

Skills name the objects 2.svg

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

Skills name the objects 3.svg

Therefore, we can name the line OQ.\overleftrightarrow{OQ}. As MM 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,N, O,O, and P.P.

Skills name the objects 4.svg

Thus, the plane can be named NOP.NOP.

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Exercise

Sketch a diagram where plane AA and plane BB intersect along PQ.\overleftrightarrow{PQ}.

Solution

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

Skills sketch the diagram 1.svg

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

Skills sketch the diagram 2.svg

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

Skills sketch the diagram 3.svg

We have now finished the sketch.

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Exercises

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