Twój koszyk jest pusty
Constructing parallel lines
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Example 1
You have a line \latex{ e } and a point \latex{ P }, which is not situated on line \latex{ e }. Plot a line \latex{ g } that is parallel to line \latex{ e } and crosses point \latex{ P }.
Solution 1
Use the fact that if line \latex{ f } is perpendicular to line \latex{ e } and line \latex{ g } is perpendicular to line \latex{ f }, then line \latex{ e } is parallel to line \latex{ g }.
Sketch:
The steps of construction:
- Plot line e and point \latex{ P }, which is not on the line.
- Construct line \latex{f}, which is perpendicular to line e and crosses point \latex{P}.
- Construct a line that is perpendicular to line \latex{f} and crosses point \latex{P}. This will be line \latex{g\parallel e}.
Construction:
\latex{P}
\latex{f}
\latex{g}
\latex{e}
Solution 2
Use the fact that the opposite sides of a rhombus are parallel.
Construct a rhombus in which one of the sides coincides with line \latex{ e } and point \latex{ P } is one of the vertices of the opposite side.
Sketch:
The steps of construction:
- Plot line \latex{e} and point \latex{P}, which is not on the line.
- Construct an arc with an arbitrarily chosen radius \latex{ a } around point \latex{P}, intersecting line \latex{e} at point \latex{Q}.
- Construct an arc with radius \latex{ a } around point \latex{Q}. Mark the intersection of the arc and line e with the letter \latex{R}.
- Construct intersecting arcs with radius \latex{ a } around points \latex{P} and \latex{R} to get point \latex{S}, the fourth vertex of the rhombus.
- Plot line \latex{PS}, which is parallel to line \latex{e}.
Construction:
\latex{P}
\latex{S}
\latex{g}
\latex{e}
\latex{Q}
\latex{R}
Discussion:
The problem has exactly one solution.
Example 2
You have line \latex{e}. Plot a line parallel to line \latex{ e } at a distance of \latex{ 2 } \latex{ cm }.
Solution
Remember that in the case of parallel lines, every point of one of the lines is found at the same distance from the other line.
If you can find two points in the half-plane defined by line e that are \latex{ 2 } \latex{ cm } away from it, then the line crossing these points is also found \latex{ 2 } \latex{ cm } away from line \latex{ e }; thus, the two lines are parallel to each other.
Sketch:
The steps of construction:
- Plot line \latex{ e } and mark points \latex{ A } and \latex{ B } on it.
- Construct lines perpendicular to points \latex{ A } and \latex{ B }.
- Construct arcs with a radius of \latex{ 2 } \latex{ cm } from points \latex{ A } and \latex{ B } that intersect the previously plotted perpendicular lines to get points \latex{ E } and \latex{ G } (and points \latex{ F } and \latex{ H } in the other half-plane).
- Plot line \latex{ g } (\latex{ h }) that crosses points \latex{ E } and \latex{ G } (\latex{ F } and \latex{ H }). ( \latex{g\parallel e} and every point of line \latex{ g } is found at a distance of \latex{ 2 } \latex{ cm } from line e.)
Construction:
\latex{g}
\latex{e}
\latex{h}
\latex{E}
\latex{A}
\latex{F}
\latex{G}
\latex{B}
\latex{H}
Discussion:
Since lines that meet the criteria can be plotted in both half-planes defined by line \latex{ e }, the problem has two solutions (\latex{ g } and \latex{ h }).
Exercises
{{exercise_number}}. There is a line \latex{ e } and a point \latex{ A }, which is not located on it. Construct a line \latex{ f } that is parallel to line \latex{ e } and crosses point \latex{ A }.
{{exercise_number}}. Construct a line parallel to a given line located at a distance of \latex{ 3 \;cm } from it. How many lines can you construct that meet the criteria?
{{exercise_number}}. Construct a square with \latex{ 45 } \latex{ mm } long sides.
{{exercise_number}}. Construct a rectangle whose sides are \latex{ 6 } and \latex{ 8 } \latex{ cm } long.
{{exercise_number}}. There is a line \latex{ e }, a point \latex{ A } on it, and a point \latex{ B }, which is not on the line. Construct a rhombus in such a way that line segment \latex{ AB } is one of its sides, while one of its diagonals is line \latex{ e }.
{{exercise_number}}. There is a line \latex{ e } and a point \latex{ P }, which is not located on it. Construct a square in such a way that point \latex{ P } is one of its vertices, while line \latex{ e } is one of its axes of symmetry.
Quiz
Funny Pepe was robbed. Detective Hound highlighted the following parts of the confessions of the three robbers.
Sticky Fingers Sam: We did not want to sell the stolen items right away, so we hid them in a suitcase and buried them \latex{250\,m} from the main road.
Loser Leo: The grassy meadow where we hid the suitcase is less than \latex{ 100 } \latex{ metres } from the railway.
Sneaky Simon: The place where we buried the suitcase is less than \latex{ 200 } \latex{ metres } from the water tower.
Where is the suitcase buried? Help Detective Hound to find it.
railway
water tower
main road
\latex{ 100 } \latex{ m }
Pepe’s house