A kosarad üres
Line reflection (reflection about a straight line) in the plane
In Figure 4 it can be seen how we can create the mirror image of a paint spot about a fold. Point P and point P' overlay each other when the paper is folded; after opening the folded paper these points are positioned so that the fold perpendicularly bisects the line segment PP'. Based on the above example we can define a planar congruent transformation: line reflection or reflection about a straight line.
DEFINITION: Straight line \latex{ t } of the plane is given. Let us assign point \latex{ P' } to every single point \latex{ P } of the plane as follows:
- if \latex{P\in t}, then \latex{ P = P’ };
- if \latex{P\notin t}, then \latex{ P’ } is the point of the plane for which it is satisfied that the perpendicular bisector of line segment \latex{ PP' } is straight line \latex{ l }.
\latex{ P' }
Figure 4
\latex{ P }
According to the assignment instruction the mirror image of point \latex{ P } about axis \latex{ l} can be constructed as follows (Figure 5):
Figure 5
\latex{ l }
\latex{ P }
\latex{ P }
\latex{ P' }
\latex{ T }
\latex{ T }
\latex{ l }
- We construct a perpendicular straight line from \latex{ P } to axis \latex{ l }; let the base point of the perpendicular straight line be point \latex{ T }.
- On the ray starting from point \latex{ P } and containing point \latex{ T } from point \latex{ T }we mark off distance \latex{ PT } on the other side of the ray; the resulting end-point will be point \latex{ P' }, the image of point \latex{ P }.
The line reflection is unambiguously defined by axis \latex{ l }, or by point \latex{ P } and its image point P', if \latex{P\neq P'}.
Example 1
Let us take triangle \latex{ ABC } and axis \latex{ l }, and let us construct the mirror image of the triangle about axis \latex{ l }.
Figure 6
\latex{ l }
\latex{ C }
\latex{ C' }
\latex{ A' }
\latex{ A }
\latex{ B }
\latex{ B' }
Solution
For the construction it is enough to construct the images of vertices \latex{ A }, \latex{ B } and \latex{ C }, these will unambiguously determine the image triangle. (Figure 6)
The properties of line reflection
- The points of the axis are fixed points (their images are themselves); there are no other fixed points, since any point not lying on the axis is separated from its image by the axis. (Figure 7)
\latex{ l }
\latex{ P = P' }
Figure 7
DEFINITION: The figures, every point of which are fixed points in the case of a certain geometric transformation, are called the fixed figures of the transformation. The axis is a fixed figure of the line reflection.
fixed figures
Figure 8
\latex{ l }
\latex{ P }
\latex{ Q }
\latex{ Q' }
\latex{ P' }
- The image of a straight line perpendicular to the axis is itself, but it is not fixed by points. The straight lines perpendicular to the axis are the invariant straight lines of the line reflection. (Figure 8)
DEFINITION: The figures, the images of which are the same as the original figure in the case of a geometric transformation, are called the invariant figures of the given transformation.
- The image of a straight line intersecting the axis obliquely (not perpendicularly) intersects the original straight line on the axis, and it includes the same angle with the axis as the original straight line (Figure 9). The figure made up by straight lines \latex{ e } and \latex{ e' } is an invariant figure of the reflection.
\latex{\alpha }
\latex{\alpha }
\latex{l }
\latex{e }
\latex{e' }
Figure 9
- The image of a straight line parallel with the axis is also parallel with the axis, and the axis bisects the bar defined by the original straight line and the image of the straight line. (Figure 10)
- Line reflection is a distance-preserving and angle-preserving transformation, i.e. any line segment and its image are of equal length, and any angle and its image are of equal measure. (Figure 11)
\latex{ l}
\latex{ e' }
\latex{ d }
\latex{ e }
\latex{ d }
Figure 10
- If in the case of a line reflection the image of point \latex{ P } is point \latex{ P' }, then in the case of the same line reflection the image of point \latex{ P' } is point \latex{ P }. So if we reflect twice in succession about the same axis, then the transformation resulting when doing these two reflections in succession assigns every single point of the plane to itself.
DEFINITION: The geometric transformations which assign every point of the plane (or the space) to itself are called identical transformations or shortly the identity.
\latex{l }
\latex{\alpha }
\latex{\alpha' }
\latex{Q' }
\latex{P' }
\latex{P }
\latex{Q }
Figure 11
- Line reflection changes the orientation of the figures, i.e. the orientations of the figure and of its mirror image are different. Therefore line reflection is commonly called as orientation-changing (or orientation-reversing) geometric transformation. For example in Figure 6 triangle \latex{ ABC } has a positive, while triangle \latex{ A'B'C' } has a negative orientation.
Exercises
{{exercise_number}}. Which of the arrows shown in the figure are the mirror images of each other about one of the drawn straight lines?
{{exercise_number}}. Two distinct points \latex{ P } and \latex{ P' } are given. Construct a straight line so that the image of point \latex{ P } is point \latex{ P' } when reflected about this straight line.
{{exercise_number}}. A triangle was reflected about the straight line with equation \latex{ y = x } in the Cartesian coordinate system. The vertices of the image triangle are \latex{ A’(3; –3) }, \latex{ B’(1; 3) }, \latex{ C’(8; 4) }. Determine the coordinates of the vertices of the original triangle.
{{exercise_number}}. In the Cartesian coordinate system the vertices of a triangle are: \latex{ A(–1; 1) }, \latex{ B(4; 3) }, \latex{ C(–3; 5) }. Reflect the triangle about
a) the \latex{x}-axis;
b) the \latex{y}-axis.
Give the coordinates of the vertices of the image triangle in both cases.
{{exercise_number}}. Construct the mirror image of a circle with \latex{ 3 } \latex{ cm } radius about a straight line not passing through the centre of the circle so that you are allowed to use only a pair of compasses during the construction.
{{exercise_number}}. Take two circles with equal radii. Construct a straight line so that the image of one of the circles is the other circle when reflected about this straight line.
{{exercise_number}}. Three straight lines are given according to the figure. Construct a point on straight line \latex{ a } the mirror image of which lies on straight line \latex{ c } when reflecting about straight line \latex{ b }.
\latex{ c }
\latex{ a }
\latex{ b }
{{exercise_number}}. An arbitrary point \latex{ P } on side \latex{ AB } of triangle \latex{ ABC } is reflected about the straight line of the interior angle bisector starting from vertex \latex{ B }, and then the resulting point \latex{ P' } is reflected about the straight line of the interior angle bisector starting from vertex \latex{ C }, and finally the newly resulting point \latex{ P’’ } is reflected about the straight line of the interior angle bisector starting from vertex \latex{ A }. What do you experience? Justify your observation.
{{exercise_number}}. Two vertices of a triangle and the straight line of the interior angle bisector starting from the third vertex are given. Construct the triangle. Examine the conditions of constructibility.
{{exercise_number}}. Points \latex{ A } and \latex{ B } can be found in one half-plane defined by straight line \latex{ e }. Construct point \latex{ P } of straight line \latex{ e } for which distance-sum \latex{ AP + PB } is the smallest possible.
{{exercise_number}}. A triangle is reflected about the straight lines of all three sides. The planar figure resulting when uniting the original triangle and the three image triangles is a triangle. What can we say about the original triangle? (Justify your observation.)
Puzzle
Two points are given in the plane. Using only a pair of compasses construct two points the distance between which is twice the distance between the two given points.
