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Applications of rotation about a point II
DEFINITION: A planar figure is rotationally symmetric, if there is point O in the plane and there is an angle \latex{\alpha} with positive direction \latex{(0º \lt \alpha\lt 360º)}, so that this figure is an invariant figure when rotated about point O through angle \latex{\alpha}.
Examples of rotationally symmetric figures:
- The circle is invariant when rotated about its centre through any angle.
- Any centrally symmetric figure is rotationally symmetric, since these are invariant when rotated about the centre of symmetry through \latex{+180^{\circ}} (or \latex{-180^{\circ}})
- Every figure which is axially symmetric about two distinct intersecting axes is rotationally symmetric. In this case the centre of the rotation which leaves the figure in its original place is the intersection point of the axes.
Example 1
Which rotations do transform a regular
- hexagon;
- \latex{n}-gon \latex{n\geq 3} to itself?
Solution (a)
The rotations about the centre of symmetry of the hexagon through \latex{ 0º } (this is the identical transformation), \latex{60^{\circ}, 120^{\circ}, 180^{\circ}, 240^{\circ}, 300^{\circ}} are all such rotations, and only these are suitable, since the image of a vertex should be a vertex when rotated. (Figure 60)
Solution (b)
In the case of a regular n-gon out of the rotations about the centre of the polygon (the centre of its circumscribed circle, the intersection point of its axes of symmetry) only those are suitable which transform a vertex to a vertex (Figure 61). In the case of these the angle of rotation can be:
\latex{0^{\circ}, \; \frac{360^{\circ}}{n},\; 2\times\frac{360^{\circ}}{n},\; 3\times\frac{360^{\circ}}{n}, \;\dots,(n-1)\times\frac{360^{\circ}}{n}}.
It can be seen from the above list that a regular \latex{ n }-gon is centrally symmetric if and only if \latex{ n } is even.
Figure 60
\latex{ 60° }
\latex{ O }
Figure 61
\latex{\frac{360^{\circ}}{n}}
Example 2
Let us give all the congruent transformations which transform a regular octagon to itself.
Solution
- identical transformation;
- \latex{ 8 } line reflections (\latex{ 4 } axes pass through the opposite vertices, \latex{ 4 } axes pass through the midpoints of the opposite sides);
- \latex{ 7 } rotations about the centre of symmetry. The angles of the rotations: \latex{45^{\circ}, 90^{\circ}, 135^{\circ}, 180^{\circ}} (point reflection), \latex{225^{\circ}, 270^{\circ}, 315^{\circ}}.
Exercises
{{exercise_number}}. Determine those rotations which transform a
- regular triangle;
- square;
- regular pentagon;
- regular dodecagon
to itself.
{{exercise_number}}. Determine those congruent transformations which transform a
- regular triangle;
- square
to itself.
{{exercise_number}}. Decide which of the following statements are true and which are false.
- There is a rotationally symmetric triangle.
- If a triangle is isosceles, then it is rotationally symmetric.
- If a quadrilateral is rotationally symmetric, then it is a square.
- The rectangle is rotationally symmetric.
- There is a kite that is rotationally symmetric.
- For any \latex{n \geq3} there is a rotationally symmetric n-gon.
- If a planar figure is rotationally symmetric, then there is a straight line about which it is axially symmetric.
- If a planar figure is rotationally symmetric, then it is also centrally symmetric.
{{exercise_number}}. Construct a regular triangle if its centroid and one of its vertices are given.
{{exercise_number}}. Construct a regular hexagon if two adjacent vertices are given.