Note: Here is a link to Answers and Comments for this lab.

Lab 7 Composition of Line Reflections II

GO TO Preferences and set the angle measure to Directed Degrees.

In each of the labs below, we will track the transformations in two ways:

Method 1.  Make an irregular polygon (a "blob") and transform the blob to see what happens to it.  Then sometimes repeat the transformation to see the result of iteration.

Method 2.  Transform a point P to its image P' and then connect P and P' with a dashed segment. Then sometimes repeat the transformation to get P'' etc and connect with segments to get a "track".

Double Reflections in Parallel Lines – Order is important

• In a new sketch, construct a line a = line OA and a parallel line b through B. Let R_A and R_B be reflections across these lines.
• Form a blob S and transform it by R_BR_A.  Draw point P and transform it to P'' = R_BR_A(P).  What kind of isometry is R_BR_A?
• Place line a to the left side of b, with the lines approximately vertical.  In this case, the point P'' should be to the RIGHT of P.  If not, you have probably found P'' = R_AR_B(P) instead of the other order.
• Measure the distance from A to line b.  Measure the distance from P to P''.  How are they related?  How can you see this geometrically.
• Now we will move a and b without changing the distance.  To do this, carefully make sure that everything is unselected.  Then select the two lines a and b (but not any points and not anything else).  Drag line of the lines.  They should both move without changing the distance.  In this case, does P' = reflection of P in a change?  Does P'' change?
• What isometry is R_AR_B and how is it related to R_BR_A?

Reflections in 3 Parallel Lines

• Add a line c to the figure, which is a line through a new point C and parallel to a and b.  Reflect P'' across OC.  You can do this with your blob also.
• What is the isometry R_CR_BR_A?
• For what line D is R_D = R_CR_BR_A?  How can you construct this line?
• If you rewrite this equation as R_CR_D  = R_BR_A, how is it related to the double-reflection = translation fact from the first experiment?

Reflections in 2 Intersecting Lines – Order is important

• Draw lines OA and OB and a blob.  Let R_A and R_B be reflections across these lines.
• Measure angle AOB.  Drag B so that this angle is about +45 degrees.
• Form a blob S and transform it by R_BR_A.  Draw point P and transform it to P'' = R_BR_A(P).
• Measure POP''. This should be about 90 degrees, not –90.  If it is the latter you found P'' = R_AR_B(P).
• What is the difference between R_BR_A and R_AR_B?  How are they related?
• What is angle AOB if R_AR_B is a point symmetry?
• It is possible to change the Selection Arrow Tool to a Rotation Tool.  Double click on O to mark it the center.  Now select both A and B with the Rotation tool and move them.  The angle AOB should stay the same.  Does the reflection of P across OA stay the same?  How about P''.

Reflections in 3 Concurrent Lines

• Add a line OC to the figure.  Reflect P'' across OC.  You can do this with your blob also.
• What is the isometry R_CR_BR_A?
• For what line D is R_D = R_CR_BR_A?  How can you construct this line?
• If you rewrite this equation as R_CR_D  = R_BR_A, how is it related to the double-reflection = rotation fact from the first experiment?

Reflections in 3 General Lines

• Draw 3 lines A1A2, B1B2, C1C2.  Let the corresponding line Reflections be R_A, R_B, R_C.
• Do this experiment two ways.  Form a blob S and reflect S 3 times to get S''' = R_CR_BR_A(S).  Hide the intermediate steps and make a tool that will take S and the 3 lines to get S'''.  If we call T = R_CR_BR_A, then we want to find T(S), TT(S), TTT(S), etc.  What pattern emerges?
• Now start with a point P and form P1 = T(P), P2 = TT(P), P3 = TTT(P), etc.  And connect P to P1 and P1 to P2, etc., with segments.  What pattern do these segments make?  What pattern to the midpoints of the segments make?
• What happens if the 3 mirror lines are move so that they are (approximately) concurrent?

GSP Definition of Translations and Glide Reflections

• Translation:  Mark Vector and Translate
• Glide Reflection.  Draw line AB.  Mark Vector AB and Mark Mirror Line AB.  For any P, translate by the marked vector to get P'and then reflect to get P''.  Hide the point P'.