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Axially symmetric figures
DEFINITION: A planar figure is axially symmetric if there is a straight line in the plane the figure reflected about which is an invariant figure.
Figure 12
\latex{ l }
\latex{ l }
\latex{ l }
The axis of the reflection for which the figure is an invariant figure is called the axis of symmetry of the figure. (Plural form: axes of symmetry.)
The circle is axially symmetric for any straight line passing through its centre.
Axially symmetric triangles
If a triangle is axially symmetric, then one of its vertices lies on the axis of symmetry, and the other two vertices are each other's mirror images when reflected about the axis.
The resulting triangle is the well-known isosceles triangle, the properties of which are resulting from the properties of the line reflection (Figure 13):
- it has two equal sides (legs);
- it has two equal angles;
- the perpendicular bisector of one of the sides (base) bisects the angle opposite.
Figure 14
\latex{ l }
\latex{ A }
\latex{ C=B' }
\latex{ B=C' }
It can be proven that any of the above three properties implies that the triangle is axially symmetric, and thus the other two properties are also fulfilled.
If all three sides of the triangle are equal, then it is a regular triangle.
A regular triangle has three axes of symmetry. (Figure 14)
Figure 15
\latex{l_1}
\latex{l_2}
\latex{l_3}
Example 1
Let us construct an isosceles triangle if the following are given: the axis of symmetry and the vertex lying on it, and two straight lines passing through the other two vertices each.
Solution
Let us find point \latex{ A } of straight line \latex{ a } the mirror image \latex{ B } of which lies on straight line \latex{ b } when reflected about \latex{ l }. (Figure 15)
The common point of \latex{ b } and the mirror image \latex{ a' } of \latex{ a } when reflected about \latex{ l} will be vertex \latex{ B }, and the mirror image of this when reflected about \latex{l } will be vertex \latex{ A }.
The common point of \latex{ b } and the mirror image \latex{ a' } of \latex{ a } when reflected about \latex{ l} will be vertex \latex{ B }, and the mirror image of this when reflected about \latex{l } will be vertex \latex{ A }.
If a' and b are parallel and do not coincide, then there is no suitable triangle. If they coincide, there are infinitely many suitable triangles, and if they intersect each other at one point, then we get an unambiguous solution.
Figure 15
\latex{C }
\latex{ l }
\latex{ B'=A }
\latex{ a }
\latex{ a' }
\latex{ b }
\latex{ B=A' }
Axially symmetric quadrilaterals
We have two options in the case of quadrilaterals:
- There is no vertex on the axis of symmetry.
- There is a vertex on the axis of symmetry.
If there is no vertex on the axis of symmetry, then two and two vertices of the quadrilateral are each other's mirror images when reflected about the axis. (Figure 16)
The quadrilateral which is axially symmetric this way is called symmetric trapezium, the axis of symmetry of which is the common perpendicular bisector of the bases.
The quadrilateral which is axially symmetric this way is called symmetric trapezium, the axis of symmetry of which is the common perpendicular bisector of the bases.
Figure 16
\latex{ B=A' }
\latex{ l }
\latex{ l }
\latex{ D=C' }
\latex{ C=D' }
\latex{ A=B' }
The axial symmetry implies the below properties of the symmetric trapezium:
- its angles on the base are of equal measure;
- its legs are of equal length;
- its diagonals intersect each other on the axis of symmetry and are of equal length.
Note: The fact that the legs of a trapezium are of equal length does not imply that it is axially symmetric. Parallelograms are isosceles trapezia, but among them only the rectangles and the rhombi are axially symmetric.
Example 2
The diagonal of a symmetric trapezium is perpendicular to the leg; the shorter base and the leg are of equal length. Let us calculate the angles of the trapezium.
Figure 17
\latex{90^{\circ}-\beta}
\latex{\beta}
\latex{ AD=DC}
\latex{ A }
\latex{ B}
\latex{ C}
\latex{ D}
Solution
If \latex{ABC\sphericalangle=\beta}, then
\latex{CAB\sphericalangle =90^{\circ}-\beta.} (Figure 17)(1)
Since \latex{AD = DC}, therefore
\latex{DAC\sphericalangle=ACD\sphericalangle}.
\latex{ACD\sphericalangle=90^{\circ}-\beta}.(2)
\latex{2(90^{\circ}-\beta)=\beta,}
which implies\latex{\beta=60^{\circ}}.
If there is a vertex on the axis of symmetry of the axially symmetric quadrilateral, then two vertices lie on the axis, and the other two vertices are each other's mirror images when reflected about the axis. (Figure 18)
The quadrilateral which is axially symmetric this way is called kite; its axis of symmetry is the straight line of one of its diagonals.
The axial symmetry implies the below properties of the kite:
- two and two adjacent sides are of equal length (earlier we gave the definition of the kite with this);
- one of its diagonals perpendicularly bisects the other diagonal;
- one of its diagonals bisects two opposite angles of the quadrilateral;
- it has two opposite angles with equal measure.
Figure 18
\latex{ l }
\latex{ A=A' }
\latex{ D=B' }
\latex{ B=D' }
\latex{ C=C' }
It can be proven that any of the first three properties imply that the quadrilateral is axially symmetric, and thus these properties are equivalent. The fourth property can also be fulfilled for quadrilaterals which are not axially symmetric. Such quadrilaterals are all those parallelograms which are not rectangles and are not rhombi.
Example 3
How many axes of symmetry do the below axially symmetric quadrilaterals have?
- concave kite
- rhombus
- rectangle
- square
Solution
The solution can be seen in Figure 19.
\latex{ 1 } axis of symmetry
\latex{ 2 } axes of symmetry
\latex{ 2 } axes of symmetry
\latex{ 4 } axes of symmetry
\latex{l}
\latex{l_1}
\latex{l_2}
\latex{l_2}
\latex{l_1}
\latex{l_1}
\latex{l_2}
\latex{l_4}
Figure 19
\latex{l_3}
Example 4
How many axes of symmetry does a regular polygon with \latex{ n } sides have?
Solution
A regular \latex{ n }-gon (regular polygon with \latex{ n } sides) has \latex{ n } axes of symmetry. In the case of an odd n these axes pass through the vertices and the midpoints of the sides opposite these vertices. (In Figure 20 the axes of symmetry of a regular nonagon can be seen.)
Figure 20
\latex{A_1}
\latex{A_2}
\latex{A_3}
\latex{A_4}
\latex{A_5}
\latex{A_6}
\latex{A_7}
\latex{A_8}
\latex{A_9}
In the case of an even \latex{ n } every second axis is the common perpendicular bisector of the opposite sides, and all other axes pass through the opposite vertices. (In Figure 21 the axes of symmetry of a regular octagon can be seen.)
It can be observed that the axes of symmetry of the regular polygons mentioned in the example all pass through one point. It can be proven that it is not only true for regular polygons, i.e. if a polygon has several axes of symmetry, then there is a point through which all of the axes pass.
Figure 21
\latex{A_1}
\latex{A_2}
\latex{A_3}
\latex{A_4}
\latex{A_5}
\latex{A_6}
\latex{A_7}
\latex{A_8}
Exercises
{{exercise_number}}. Decide which of the below statements are true and which are false.
- Every triangle is axially symmetric.
- There is an axially symmetric triangle.
- If a triangle has an axis of symmetry, then its sides are of equal length.
- If a triangle has an axis of symmetry, then it has two sides of equal length.
- If a quadrilateral is axially symmetric, then it has a vertex which lies on the axis of symmetry.
- If a quadrilateral is axially symmetric, then it has two angles of equal measure.
- The axially symmetric quadrilaterals are convex.
- There is an axially symmetric quadrilateral, which can be split into two axially symmetric triangles.
- Any regular polygon has at least three axes of symmetry.
- Only the regular polygons are axially symmetric.
- Every regular polygon has a diagonal which is an axis of symmetry.
{{exercise_number}}. Construct a kite if three vertices are given so that two of these lie on the axis of symmetry.
{{exercise_number}}. Construct a symmetric trapezium if the following are given: the axis of symmetry and two vertices which are on one side of the axis of symmetry.
{{exercise_number}}. Construct a kite if the following are given: two vertices lying on its axis of symmetry and two straight lines on each of which one vertex lies. Examine the conditions of constructibility.
{{exercise_number}}. Construct a rhombus if two axes of symmetry and two points on one of its sides are given. Examine the conditions of constructibility.
{{exercise_number}}. Determine the coordinates of the vertices of the square with \latex{ 2 }-unit-long sides in the coordinate system which is symmetric about both the \latex{ x }-axis and the \latex{ y }-axis. How many solutions does the exercise have?
{{exercise_number}}. Prove that if a triangle has two axes of symmetry, then it has three of them.
{{exercise_number}}. On the internet search for such artworks and photos on which axially symmetric figures appear.