SYMMETRY
Geometry
 
1. LINE SYMMETRY 
Symmetry about the line r transforms point A into A'. Therefore, line r is the perpendicular bisector of segment AA´. Symmetry respects distances but does not respect the direction of the angles (clockwise or anticlockwise). 
1.- Look at the order of points in the yellow triangle and the turquoise one, which is its reflection about the line of symmetry. Are they the same or different? 

2.- Move point B with the mouse until it is lined up exactly with points A and C. Look at how the symmetry changes the order of the points.

3.- Move points A, B and C so that the triangle symmetrical to it fits over the original one exactly. For example, place C on the line of symmetry and points A and B on B' and A' respectively. Note that triangle ABC is an isosceles triangle and that it fits exactly over its reflection A´B´C´. In this case we can say that triangle ABC has one line of symmetry. How many lines of symmetry does an equilateral triangle have?
2. LINES OF SYMMETRY 
If a shape fits exactly over another, which is symmetrical to it, about a line, we can say that it has a line of symmetry. In the Descartes window there are four points which form a rectangle and a line of symmetry r, which can be moved to the left or to the right.
4.- Move the line of symmetry until the reflection of the rectangle ABCD fits exactly over itself. At this point we can say that r is the line of symmetry of the rectangle. Where does it pass through? Do you know if it has another line of symmetry? 

5.-Click on the Init button and construct an isosceles triangle whose base is 4 by joining together points A and B of the rectangle. Use the method explained above to find its line of symmetry, if it has one, and which points this line passes through. Does it have another line of symmetry? What about if it were an equilateral triangle?
3. SYMMETRY ON THE CARTESIAN PLANE 
Symmetry about the axes on the Cartesian plane is expressed in a very straight-forward way. If we call the Y-axis r and the X-axis s the reflections about these lines of symmetry for point A are referred to as Ar and As.

6.- Find the reflection of the following points about lines r and s: A(1,1), B(-2,3), C(2,-1), D(-2,-3).

7.- Given any point with coordinates (x,y), work out its new coordinates after a reflection about each axis.

8.- Work out the coordinates of the squares symmetrical to the square A(1,1), B(1,4), C(4,4) and D(4,1) about lines r and s and draw the reflections in your exercise book.

 
4. COMBINations of SYMMETRY 
If two transformations involving symmetry are applied the following can occur:
  • The same transformation involving symmetry is applied twice, which is the same as an identity.
  • If two transformations involving symmetry occur and the lines are parallel, when they are combined the result is a translation where the shape moves double the distance between the two lines.
  • When the lines intersect the result is a rotation about the point where the lines intersect through an angle which is double the angle formed by the axes.

We are going to focus on the first two cases in this Descartes window as we have two parallel lines r and s, which transform triangle ABC into A´B´C´ and A´´B´´C´´ respectively. 

8.-Move the vertices of the yellow triangle and watch how the shape and size of the corresponding symmetrical triangles changes too.

9.- Drag the lines in both directions to show that the relation between the distance of the movement and the distance between the lines is satisfied.

10.- Drag line s and place it on top of line r. Look at how triangle ABC fits exactly over A´´B´´C´´.

11.-Click on the Init button and draw a similar example to the one given in the window in your exercise book. Check that the total translation is equal to double the distance between the two lines.

12.- Find out what would happen if s was the first line of symmetry and then r. Would you get the same end result?

 
In this Descartes window we are going to see an example of when the lines are not parallel. We shall see how the combination of transformations is a rotation through an angle which is double the angle formed between the two lines.

13.- Move point r and s in the Descartes window to create different transformations of symmetry and look carefully at the size of the angle of rotation which is produced.
       
           
  Miguel García Reyes
 
Spanish Ministry of Education. Year 2001
 
 

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