Line Reflection and Line Symmetry

Definition of Line Reflection:

• Draw a line m and a point A in Sketchpad.  Reflect the point across the line to get point A'. Construct segment AA'.  Drag A around to see how AA' varies.  How is the line related to A?
• Based on your observations, write a good definition of line reflection of a point A in a line m.  Be sure your definition includes the case when A is on m.

Your Definition: ________________________________________________

Line Symmetry and a Line of Symmetry

Let m be a line and S be some figure.  If the reflection S' of S in line m is the same as S, then we say that S has line symmetry  with line of symmetry m.

Triangles with a Line of Symmetry

• Draw a triangle ABC and a line m.  Reflect the triangle across line m and color the reflected triangle a color different from that of ABC. 
• The line m is a line of symmetry of ABC if the triangles are superimposed. Drag the vertices A, B, C around until you find a position where the two triangles are superimposed.

Conjecture:  If the triangle ABC has a line of symmetry, what kind of triangle is ABC?  How is the line related to ABC?

WriteYour Conjecture: _______________________ …

Conjecture:  If the triangle ABC has two lines of symmetry, what kind of triangle is ABC?  How are the lines related to ABC?

WriteYour Conjecture: _______________________ …

If you wanted to prove your conjecture, you could try the following approach. Consider several possible cases.

Answer:

• Suppose NONE of the vertices of triangle ABC is on line m, then m CANNOT be a line of symmetry of ABC.
  • Experiment with some examples of this case. Why can m not be a line of symmetry? _________
• Suppose ONE of the vertices of triangle ABC is on line m, then m CAN be a line of symmetry of ABC.
  • What must be true about B and C if vertex A is on line m, which is a line of symmetry?__________ Why?
• Suppose TWO of the vertices of triangle ABC are on line m, then m CANNOT be a line of symmetry of ABC.
  • Experiment to see the reason. Why? _______

Quadrilaterals with a Line of Symmetry

• Inspired by your experience with triangles, explain exactly what quadrilaterals have a line of symmetry.
• Sketch all the types of quadrilaterals with a line of symmetry and tell what they are.

• Sketch a pentagon with a line of symmetry

Circles, Equilateral triangles, rectangles

• Use Sketchpad to investigate what are the lines of symmetry of circles, equilateral triangles and rectangles.  How many lines of symmetry in each case?
  • Circles?____________
  • Equilateral Triangles?____________
  • Rectangles?____________
• How many lines of symmetry does a square have? ________
  

Multiple Mirrors and Kaleidoscopes

Start with the center P and point 0. Construct a square as in the figure, divided into 8 congruent right triangles as in this figure.  Number the endpoints of the segments as shown.  Then draw freehand a"blob" (it is a general polygon interior) and place it as shown.

Goal: To understand the effect of multiple reflections and see how multiple reflections explain the view into a kaleidoscope.

Important:  Use File > Document Options to make a 4 additional pages in your GSP document that are each a copy of your first page with the square above.  You will need it later.

We will be studying the effects of reflecting the blob, which we call S.  Here is the notation we will use.

If n is a number, Rn means reflection in line Pn.  For example R0 is reflection in line P0, R1 is reflection in line P1, etc.

Composition:  For any object T, RjRi(T) means FIRST reflect by Ri and SECOND reflect by Rj.

Experiment 1.  One the first page, fill in a pattern by reflecting S and its images by various Ri to get as many blobs as possible.  What are the lines of symmetry of the result?

• Drag the blob around and see how the pattern of 8 blobs moves in two groups of 4.
• Drag the point 0, leaving P fixed, and again see what happens to the original blob and 3 of the other images.

Experiment 2.  Start afresh with your copy on page 2.  Fill in the pattern by reflecting S and the images of S (and their images, etc.) but this time only use R1 and R0 over and over in any order you want.

• How does this compare with the results of Experiment 1, when you could reflect across any of the Ri? What are the lines of symmetry of the result?
• Make a note of which blob is R1(R0(S)).

Experiment 3.  Start afresh with your copy on page 3.  Fill in the pattern by reflecting S and its images only using R2 and R3 over and over in any order you want.

• How does this compare with the results of Experiments 1 and 2?
• Make a note of which blob is R3(R2(S)). How does this compare with R1(R0(S))?

Experiment 4.  Start afresh with your copy on page 4.  Fill in the pattern by reflecting S and its images only using R2 and R0 over and over in any order you want.

• What are the lines of symmetry of the result?
• How does this compare with the results of Experiments 1, 2 and 3?

Application to Kaleidoscopes

If one takes two hinged mirrors that meet at an angle of 45 degrees, the kaleidoscopic pattern always has 8 congruent parts and 4 lines of symmetry.

If one takes two hinged mirrors that meet at an angle of 90 degrees, the kaleidoscopic pattern always has 4 congruent parts and 2 lines of symmetry.

Double Line Reflection in General

Experiment 5.  In any of your sketches, draw a new point Q in the plane and reflect by R0 to get Q' and reflect Q' by R1 to get Q''. 

• Measure angle QPQ'' and move Q around to see how the angle changes.
• What can you say about this angle?
• Look at the triangles QPQ' and Q'PQ''. Explain why both are isosceles.
• Notice that the mirror lines are lines of symmetry of these triangles. Can you see why the angle QPQ'' is what it is, wherever Q is located? Explain.

Experiment 6.  Start afresh in a blank sketch, and draw two lines PA and PB.  Repeat experiment 5 with reflections in these lines. (Specifically, Q' = reflection of Q in PA and Q'' = reflection of Q' in PB.) Measure the angles with Sketchpad. Look for the relation between angle APB and angle QPQ''.

Big Questions

• How is angle QPQ'' related to angle APB?
• Why is this relationship true? Is this easy to see in at least some locations of P?
• What kind of transformation is the composition of line reflections in intersecting lines?