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Points and lines

Points and lines are basic elements of planimetrics.

Point has no size (no width, no length, no depth) and is shown by a dot. It’s usually named with a capital letter.

Line is a collection of points arranged in a straight path that’s endless in both directions. Therefore it has one dimension: length.
A line can be identified in two ways: either by two points that are on the line, or by a lowercase letter. It has no ending on both sides.

This is line $AB$ or$\overleftrightarrow{AB}$

Also, we could call it line $a$.

Ray is a line that has an endpoint on one side and continues off to infinity on the other side.  We can name a ray using its starting point and one other point that is on the ray, or we can use a lowercase letter.

This is ray $AB$, $\overrightarrow{AB}$, or simply ray $a$.

Line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.

This is line segment $AB$ or simply $\overline{AB}$.

There are $3$ tipes of line segment:

  • closed line segment – includes both endpoints
  • open line segment – excludes both endpoints
  • half-open line segment – includes only one endpoint (doesn’t matter which one)

Relationships between a point and a line

  • $A \in p$
    There are two ways to read this: line $p$ goes through the point $A$ or point $A$ lies on the line $p$

  • $A \notin p$
    Meaning: line $p$ doesn’t go through the point $A$ or point $A$ doesn’t lie on the line $p$

  • $p \bigcap q = {A}$
    Meaning: lines $p$ and $q$ intersect on point $T$ or point $T$ is an intersection of lines $p$ and $q$.

  • Points $A$ and $B$ lie on the same side of line $p$ from the point $C$

  • $A$ and $B$ lie on the opposite sides of line $p$ from the point $C$, or point $C$ is between points $A$ and $B$

  • Points $A$ and $B$ lie on the same side of line $p$

  • $A$ and $B$ lie on the opposite sides of line $p$

 

$\Longrightarrow$ relationships between ray (line segment) and point are the same as above.

 

Relationship between two lines

  • Intersecting lines
    A pair of lines are intersecting if they have a common point, also called as point of intersection.

    $\quad$ Perpendicular lines
    A special type of intersecting lines where the angle of intersection is a right angle.
  • Parallel lines
    Two lines are parallel if they lie in the same plane and do not intersect when extended on either side.
    We say “$p$ is parallel to $q$” and write $p||q$.

Euclid’s fifth postulate (parallel postulate)
“In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.”

Note that the distance between two parallel lines is the same everywhere.

$\Longrightarrow$ relationships between ray (line segment) and point are the same as above.

 

Line segments worksheets

  Constructing line segments (173.2 KiB, 477 hits)

  Constructing angles (118.4 KiB, 648 hits)

  Construction of angle bisectors (438.2 KiB, 624 hits)

  Measuring lines in centimeters (105.0 KiB, 670 hits)

  Measuring lines in millimeters (73.8 KiB, 669 hits)