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Point reflection in the plane
Example 1
Let us take a triangle in the first quadrant of the Cartesian coordinate system; let its vertices be \latex{A(6; 4), B(2; 3), C(5; 1)}. Let us reflect this triangle about the y-axis, and then let us reflect the resulting triangle \latex{A’B’C’} about the x-axis.
Solution
When connecting vertices \latex{A’’, B’’} and \latex{C’’} of the triangle resulting after the second reflection with vertices A, B and C respectively, we experience that the common midpoint of line segments \latex{AA’’, BB’’, CC’’} is the origin. (Figure 22)
Figure 22
\latex{ C (5;1) }
\latex{ A' (-6;4) }
\latex{ A'' (-6;-4) }
\latex{ y}
\latex{ C'' (-5;-1) }
\latex{ C' (-5;1) }
\latex{ B' (-2;3) }
\latex{ B (2;3) }
\latex{ A (6;4) }
\latex{ B'' (-2;-3) }
\latex{ x}
\latex{ O}
Based on this we can define a new congruent transformation in the plane, the point reflection (reflection in a point).
DEFINITION: Point \latex{ O } of the plane is given. Let us assign point \latex{ P' } to every single point \latex{ P } of the plane as follows:
- we assign \latex{ O } to itself, i.e. \latex{ O = O’ };
- if \latex{P \neq O}, then \latex{ P’ } is the point of the plane for which it is satisfied that the midpoint of line segment \latex{ PP’ } is point \latex{ O }.
Point reflection can be defined in the space in the same way.
According to the assignment instruction the mirror image\latex{ P' } of point \latex{ P } with centre \latex{ O } can be constructed by simply copying the line segment on the ray starting from point \latex{ P } and containing point \latex{ O }.
Point reflection is unambiguously defined by point \latex{ O }, or by a point \latex{ P } (different from point O) and its image point \latex{ P' }.
Example 2
Let us take triangle \latex{ ABC } and point \latex{ O }, and let us construct the mirror image of the triangle in point \latex{ O }.
Figure 23
\latex{A' }
\latex{ A }
\latex{ B }
\latex{ O }
\latex{C }
\latex{C' }
\latex{B' }
Solution
The mirror images of the vertices of the triangle unambiguously define the image triangle. (Figure 23)
The properties of point reflection
- The sole fixed point of the transformation is centre \latex{ O } of reflection.
- The straight lines passing through centre \latex{ O } of reflection and only these straight lines are the invariant straight lines of the point reflection. (Figure 24)
- The image of a straight line not passing through centre \latex{ O } of the reflection is a straight line parallel with the original one and not passing through centre \latex{ O }. (Figure 25)
- Point reflection is a distance-preserving and angle-preserving transformation. (Figure 26)
Figure 24
\latex{ O }
\latex{ P }
\latex{ P' }
- If in the case of a point reflection the image of point \latex{ P } is point \latex{ P' }, then in the case of the same point reflection the image of point \latex{ P' } is point \latex{ P }. So when doing two point reflections with the same centre in succession results in the identical transformation.
- Point reflection can be produced by doing two line reflections with perpendicular axes in succession. The centre of the reflection is the intersection point of the two axes. (Figure 27)
(It can be proven that this production is independent of the direction of the axes, i.e. one of the axes can freely be chosen and the other axis will be perpendicular to it.)
- The previous property (using that line reflection is an orientation-changing geometric transformation) implies that point reflection is an orientation-preserving geometric transformation.
the properties of point reflection
Figure 25
\latex{ e'}
\latex{ e }
\latex{ Q }
\latex{ P}
\latex{ O}
\latex{ Q'}
\latex{ P'}
\latex{O}
\latex{\alpha'}
\latex{AB=A'B'}
\latex{AB=A'B'}
\latex{\alpha=\alpha'}
Figure 26
\latex{\alpha}
\latex{A}
\latex{B}
\latex{B}
\latex{A}
Figure 27
\latex{A}
\latex{t_2}
\latex{t_1}
\latex{A''}
\latex{B''}
\latex{B}
\latex{C}
\latex{C''}
\latex{O}
\latex{C'}
\latex{B'}
\latex{A'}
Exercises
{{exercise_number}}. Which of the arrows shown in the figure are the mirror images of each other when reflected in a centre of any of \latex{O_1, O_2, O_3, O_4}?
\latex{O_1}
\latex{O_2}
\latex{O_3}
\latex{O_4}
{{exercise_number}}. Two distinct points \latex{ A } and \latex{ B } are given in the plane. Construct the centre of the point reflection which transforms point \latex{ A } to point \latex{ B }. What will be the image of point \latex{ B } in the case of this reflection?
{{exercise_number}}. Two circles with equal radii are given in the plane. Construct a point so that the image of one of the circles is the other circle when reflected in this point.
{{exercise_number}}. In the Cartesian coordinate system the vertices of a triangle are: \latex{A(–1; 1), B(4; 3), C(–3; 5)}. Reflect the triangle in
- the origin;
- the point \latex{(1; 0)};
- the point \latex{(2; –2)}.
Give the coordinates of the vertices of the image triangle in all three cases.
{{exercise_number}}. A triangle was reflected in the point \latex{(–3; 2)} in the Cartesian coordinate system. The vertices of the image triangle are: \latex{A’(–3; 3), B’(1; 3), C’(8; 4)}. Determine the coordinates of the vertices of the original triangle.
{{exercise_number}}. Reflect a regular triangle with \latex{ 4\, cm } long sides in the centre of its circumscribed circle. What planar figure is defined by
- the common part of the original triangle and of the image triangle;
- the union of the original triangle and the image triangle?
{{exercise_number}}. Take two non-parallel straight lines and a point not lying on them. Construct one point on each of the straight lines which will be each other's images when reflected in the given point.
{{exercise_number}}. Reflect a triangle in the midpoints of all three sides. What planar figure is defined by the union of the original triangle and the three image triangles? (Justify your observation.)
{{exercise_number}}. Take two intersecting circles with different radii. Construct a secant line passing through one of the intersection points from which the two circles are cutting chords of equal length.
{{exercise_number}}. Take a convex angle and point \latex{ P } in the angular domain. Construct a straight line passing through point \latex{ P } the line segment of which inside the angular domain is bisected by point \latex{ P }.
Puzzle
Kate and Dan are placing identical coins in turns on a round table so that none of the coins are overlapping. The player who can place a coin last wins the game. Kate places the first coin on the table. How shall she play if she wants to win?
