Vaša košarica je prazna
The position of lines
Lines can be grouped in different ways depending on their position in space.
they have one common point
they have no common points
(or have an infinite number of common points)
(or have an infinite number of common points)
they have no common points
intersecting
lines
parallel
lines
skew
lines
coplanar lines (lie on the same plane)
not coplanar
\latex{ e }
\latex{ f }
\latex{ M }
\latex{ e }
\latex{ f }
\latex{ e }
\latex{ f }
Parallel lines
Two lines are parallel if they are coplanar and do not intersect.
They are denoted as the following: \latex{ e } || \latex{ f }.
They are denoted as the following: \latex{ e } || \latex{ f }.
Example 1
There is a line denoted by \latex{ e } which does not pass through point \latex{ P }. Draw line \latex{ f } in such a way that it passes through point \latex{ P } and is parallel to line \latex{ e }.
Solution
- Place a triangular ruler on line \latex{ e }.
- Place a straight ruler next to the triangular ruler as shown.
- Slide another triangular ruler along the straight one until it reaches point \latex{ P }. Draw line \latex{ f }.
- In this way, line \latex{ f } will be parallel to line \latex{ e } and pass through point \latex{ P }.
\latex{ P }
\latex{ e }
\latex{ P }
\latex{ e }
\latex{ P }
\latex{ e }
\latex{ P }
\latex{ e }
\latex{ f }
Intersecting lines
When two lines intersect, they divide the plane into four parts. The opposite pairs of these parts are congruent. If the intersecting lines divide the plane in such a way that all four parts are congruent, then the lines are perpendicular to each other.
e
f
two parts are congruent
e
f
four parts are congruent
Two intersecting lines are perpendicular if they divide the plane into four congruent parts. They are denoted as the following: \latex{ e } ⊥ \latex{ f } (in text) and ⦝, ⦜ (on a drawing)
The two shorter sides of a triangular ruler are perpendicular to each other. If you draw the outline of a triangular ruler, you get a right-angled triangle. The two shorter sides of such a triangle are called the legs and the longest side is called the hypotenuse.
leg
hypotenuse
leg
hypotenuse
Example 2
There is a line denoted by \latex{ e } which does not pass through point \latex{ P }. Draw line \latex{ f } in such a way that it passes through point \latex{ P } and is perpendicular to line \latex{ e }.
Solution
- Place a straight ruler next to the line \latex{ e }.
- Then place any leg of a triangular ruler against it.
- Slide the triangular ruler until its other leg reaches point \latex{ P }. Draw ray \latex{ f } from line \latex{ e } to point \latex{ P }.
- Extend the ray to a line.
- In this way, line \latex{ f } will be perpendicular to line \latex{ e } and pass through point \latex{ P }.
\latex{ e }
\latex{ P }
\latex{ P }
\latex{ e }
\latex{ P }
\latex{ e }
\latex{ P }
\latex{ e }
\latex{ f }
The distance between a point and a line, or between two lines
Point \latex{ P } can be connected to line \latex{ e } with an infinite number of segments. However, the perpendicular one is the shortest of all. The length of the shortest segment drawn from a point to a line is known as the distance between the point and the line.
\latex{ P }
\latex{ R }
\latex{ e }
\latex{ d }
Example 3
Draw points that are \latex{ 1 \,cm } away from a given line.
Solution
- Draw a line denoted by \latex{ e } and mark an arbitrary point on it denoted by \latex{ T }. Draw a line perpendicular to \latex{ e } through point \latex{ T }.
- Set your compass to \latex{ 1\,cm } on the ruler and draw an arc from point \latex{ T }.
- Repeat this process at other arbitrary points.
- Connect the intersections of the perpendicular lines and the arcs.
\latex{ g }
\latex{ e }
\latex{ f }
\latex{ T }
Connecting the points creates two lines, both \latex{ 1\,cm } away from line \latex{ e }.
These are parallel to line \latex{ e }: \latex{ e } || \latex{ g } and \latex{ e } || \latex{ f }.
These are parallel to line \latex{ e }: \latex{ e } || \latex{ g } and \latex{ e } || \latex{ f }.
210392
When solving Exercise 3, you may have noticed that when two lines are parallel, the distance from any point on one line to the other is equal to the distance between lines \latex{ e } and \latex{ f } (\latex{ e } and \latex{ g }), which in this case, is exactly \latex{ 1 \,cm }.
The distance between two lines is the length of the segment from any point on
line \latex{ e } to line \latex{ f }.
line \latex{ e } to line \latex{ f }.
The distance between two intersecting lines is \latex{ 0 }.
\latex{ e }
\latex{ P }
\latex{ Q }
\latex{ R }
\latex{ f }
\latex{ P' }
\latex{ Q' }
\latex{ R' }
\latex{ d }
\latex{ d }
\latex{ d }
\latex{ e }
\latex{ f }
\latex{ d = 0 }
\latex{ P }
Exercises
{{exercise_number}}. Look for streets on the map that are
a) perpendicular;
b) parallel to each other.
b) parallel to each other.
{{exercise_number}}. Draw lines according to the symbols in
the table.
the table.
\latex{ e }
\latex{ f }
\latex{ g }
\latex{ e }
\latex{ f }
\latex{ g }
||
||
||
||
||
⊥
⊥
⊥
⊥
{{exercise_number}}. Line \latex{ e } represents one edge of a cube. Look for other edges that
a) are parallel with line \latex{ e };
b) intersect line \latex{ e };
c) are skew to line \latex{ e }. (→)
b) intersect line \latex{ e };
c) are skew to line \latex{ e }. (→)
\latex{ f }
\latex{ k }
\latex{ g }
\latex{ i }
\latex{ l }
\latex{ a }
\latex{ d }
\latex{ b }
\latex{ h }
\latex{ c }
\latex{ j }
\latex{ e }
{{exercise_number}}. Look for pairs of parallel and perpendicular lines in the following images.
a)
\latex{ f }
\latex{ e }
\latex{ h }
\latex{ g }
\latex{ i }
\latex{ j }
b)
\latex{ f }
\latex{ e }
\latex{ g }
\latex{ h }
\latex{ i }
c)
\latex{ e }
\latex{ f }
\latex{ g }
\latex{ h }
\latex{ i }
{{exercise_number}}. Draw line \latex{ e }, then
a) draw line \latex{ f } perpendicular to line \latex{ e }. Then draw another line denoted by \latex{ g } perpendicular to line \latex{ f }. What is the relationship between lines \latex{ e } and \latex{ g }?
b) draw line \latex{ f } \latex{ 2\,cm } away and parallel to line \latex{ e }. Then draw another line denoted by \latex{ f } perpendicular to line \latex{ g }. What is the relationship between lines \latex{ e } and \latex{ g }?
c) draw line \latex{ f } parallel to line \latex{ e }. Then draw another line denoted by \latex{ g } parallel to line \latex{ f }. What is the relationship between lines \latex{ e } and \latex{ g }?
b) draw line \latex{ f } \latex{ 2\,cm } away and parallel to line \latex{ e }. Then draw another line denoted by \latex{ f } perpendicular to line \latex{ g }. What is the relationship between lines \latex{ e } and \latex{ g }?
c) draw line \latex{ f } parallel to line \latex{ e }. Then draw another line denoted by \latex{ g } parallel to line \latex{ f }. What is the relationship between lines \latex{ e } and \latex{ g }?
Quiz
Without lifting your pencil from the paper, draw a line of four straight segments through all nine points.
