Jūsu grozs ir tukšs
Constructing perpendicular lines
Example 1
Construct the perpendicular bisector of a \latex{ 6 \,cm } long line segment.
Solution
Every point located at the same distance from the ends of the line segment is on the perpendicular bisector. So, construct two points located at the same distance from the ends of the line segment. The line crossing these points is the perpendicular bisector of the line segment.
The steps of construction:
- Draw line segment \latex{ AB }.
- Construct two arcs with the same radii that intersect each other in two points around points \latex{ A } and \latex{ B }.
- The line passing through points of intersection \latex{ E } and \latex{ F } is the perpendicular bisector of line segment \latex{ AB }.
Sketch:
The diagonals of a rhombus intersect at a right angle.
\latex{ E }
\latex{ A }
\latex{ B }
\latex{ F }
\latex{\text{\textcolor{c20100}{Construction:}}}
Discussion:
The task has exactly one solution.
Constructing a perpendicular line at a given point \latex{ P } on a line
Example 2
There is a line \latex{ e } with a point \latex{ P } on it. Construct a line \latex{ g } that is perpendicular to line \latex{ e } and crosses point \latex{ P }.
Solution
The steps of construction are similar to those of a line segment's perpendicular bisector. Mark an arbitrary segment \latex{ AB } with midpoint \latex{ P } on line e and construct the perpendicular bisector of line segment \latex{ AB }.
Sketch:
The steps of construction:
- Draw line \latex{ e } and mark a point \latex{ P } on it.
- Draw an arc with an arbitrary radius around point \latex{ P }, intersecting line \latex{ e } twice, thus marking points \latex{ A } and \latex{ B }.
- Draw intersecting arcs longer than line segment \latex{ AP } around points \latex{ A } and \latex{ B }.
- Construct line \latex{ g } that crosses one of the points of intersection of the arcs \latex{ (M) } and point \latex{ P }.
\latex{g}
\latex{e}
Lines \latex{ g } and \latex{ e } are each others' axes of symmetry.
\latex{\text{\textcolor{c20100}{Construction:}}}
\latex{ M }
\latex{ A }
\latex{ B }
\latex{ P }
\latex{ g }
\latex{ e }
Discussion:
The task has exactly one solution.
Constructing a perpendicular line from point P, which is not located on the line
Example 3
There is a line \latex{ e } and a point \latex{ P }, which is not located on the line. Construct line \latex{ g }, perpendicular to line e and crossing point \latex{ P }.
Solution
Mark a line segment \latex{ AB } on line \latex{ e }, whose endpoints are located at the same distance from point \latex{ P }. The perpendicular bisector of line segment \latex{ AB } is line \latex{ g }.
The steps of construction:
- Draw line \latex{ e } and mark point \latex{ P }.
- Draw an arc with centre \latex{ P } that intersects line \latex{ e } at two points. Mark the intersections with the letters \latex{ A } and \latex{ B }.
- Draw intersecting arcs with the same radius with centres \latex{ A } and \latex{ B }. The intersection is \latex{ M }.
- Draw line \latex{ MP }, which is perpendicular to line \latex{ e }.
Sketch:
\latex{\text{\textcolor{c20100}{Construction:}}}
\latex{ P }
\latex{ A }
\latex{ M }
\latex{ B }
\latex{ g }
\latex{ e }
Discussion:
The task has exactly one solution.
Constructing points located at a given distance from a line
Example 4
There is a line \latex{ e }. Construct point \latex{ P } located at a distance of \latex{ 2 \;cm } from line \latex{ e }.
Solution
The distance between a point and a line equals the length of a line segment that crosses the given point and is perpendicular to the given line. Therefore, point \latex{ P } is found on a line perpendicular to line \latex{ e } that crosses point \latex{ T }. Point \latex{ P } is found at a distance of \latex{ 2\,cm } from point \latex{ T }.
Sketch:
The steps of construction:
- Mark an arbitrary point \latex{ T } on line \latex{ e }.
- Construct a perpendicular line through point \latex{ T }, as in example 2.
- Measure \latex{ 2 \;cm } from point \latex{ T } on the perpendicular line in both directions. Points P1 and P2 are located at a distance of \latex{ 2 \;cm } from line \latex{ e }.
\latex{\text{\textcolor{c20100}{Construction:}}}
\latex{ P_1 }
\latex{ A }
\latex{ P_2 }
\latex{ T }
\latex{ B }
\latex{ e }
Discussion:
By repeating the same steps, you can construct an infinite number of points that are located at distances of \latex{ 2 \;cm } from line \latex{ e }.
\latex{ 2\,cm }
\latex{ 2\,cm }
\latex{ 2\,cm }
\latex{ 2\,cm }
\latex{ f }
\latex{ e }
\latex{ g }
\latex{f \parallel e\parallel g}
Exercises
{{exercise_number}}. Construct three line segments with different lengths and positions.
- Construct their midpoints.
- Construct the lines that quadrisect the line segments.
{{exercise_number}}. Construct and colour the following parts of an arbitrary \latex{ AB } line segment.
- \latex{\frac{1}{2}}
- \latex{\frac{3}{4}}
- \latex{\frac{5}{8}}
- \latex{\frac{5}{4}}
{{exercise_number}}.Draw an arbitrary line \latex{ g } and a point \latex{ G } that is not located on the line. Construct the distance between point \latex{ G } and line \latex{ g }.
{{exercise_number}}. Construct the diameter of an arbitrary circle. Construct the diameter perpendicular to the previously constructed one. Connect the endpoints of the diameters. What shape is the resulting quadrilateral?
{{exercise_number}}. Draw an \latex{ ABC } triangle and construct the perpendicular bisectors of the triangle's sides. What do you notice?
{{exercise_number}}. Perform the following construction.
- Draw a line \latex{ e } and mark an arbitrary point \latex{ P } on it.
- Construct a line \latex{ f } perpendicular to line \latex{ e } that crosses point \latex{ P }.
- Measure \latex{ 4 \,cm } on lines \latex{ e } and \latex{ f } from point \latex{ P } in one direction. Mark the point on line \latex{ e } with the letter \latex{ E } and that on line \latex{ f } with the letter \latex{ F }.
- Construct intersecting arcs with centres \latex{ E } and \latex{ F } with a radius of \latex{ 4 \,cm }. Mark their intersection with the letter \latex{ Q }.
How many axes of symmetry does the \latex{ PEQF } quadrilateral have?
{{exercise_number}}. A fox and a wolf cannot decide which one of them should eat the rabbit. They agree that whoever is closer to the rabbit can eat it. Where should the rabbit go so neither the fox nor the wolf can eat it? Construct the path the rabbit should take to escape.
wolf
fox
Quiz
A toy shop is found at equal distances from the houses in the image. Construct the location of the toy shop.
