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: ________________________
____ ….
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. Drag the vertices A, B, C around until 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 a 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:
Answer:
Suppose NONE of the vertices of triangle ABC is on line m, then m CANNOT be a line of symmetry of ABC. Why?
Suppose ONE of the vertices of triangle ABC is on line m, then m CAN be a line of symmetry of ABC. When? Why?
Suppose TWO of the vertices of triangle ABC are on line m, then m CANNOT be a line of symmetry of ABC. When? 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 cases of quadrilaterals with a line of symmetry and tell what they are.
Sketch a pentagon with a line of symmetry
Use Sketchpad to investigate what are the line of symmetry of circles, equilateral triangles and rectangles. How many lines of symmetry?
How many lines of symmetry does a square have?
Construct a square and divide it into 8 congruent right triangles as in this figure. Number the endpoints of the segments as shown. Then draw freehand and "blob" (it is a polygon interior) and place it as show.
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Now, 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. Fill in the 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?
Experiment 2. Start afresh with your copy. Fill in the pattern by reflecting S and its images only using R1 and R0 over and over in any order you want. What are the lines of symmetry of the result?
Experiment 3. Start afresh with your copy. Fill in the pattern by reflecting S and its images only using R2 and R3 over and over in any order you want. What are the lines of symmetry of the result?
Experiment 4. Start afresh with your copy. 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?
Experiment 5. Take 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.
Experiment 6. Start afresh 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.) Look for the relation between angle APB and angle QPQ''.
Report Question: What kind of transformation is the composition of line reflections in intersecting lines?