We have looked at how the four different transformations in the Sketchpad Transform menu have different effects on size, shape, and orientation. Remember the advice to Alice! The idea of this lab is to see the power of combining two successive reflections in the plane.

1. Draw any weird shape as a polygon interior. (Use 20 or so points to outline the shape, select the points in the order that they trace the shape you want, and choose Polygon Interior, from the Construct menu.) Call the shape W, for weird. Now, to investigate what happens to W as it is transformed in the plane.

a. Make a line L1 and a point P not on L1. Construct a line L2 that is parallel to L1 and that passes through P.

b. Mark L1 as a mirror and reflect W in L1 to get W'; then mark L2 as a mirror and reflect W' in L2 to get W''

The rest of #1 is to investigate the effect on W of two successive reflections in two parallel lines and to form a conjecture about this combination of reflections. Be sure to include in your investigation:

(i) Dragging a point of W

(ii) Dragging one of the parallel lines

c. State a conjecture or conjectures about the effect on W of two successive reflections in two parallel lines. Of course, we know that a reflection changes neither size nor shape. Try to be explicit about what transformation is the same as this combination of reflections.

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2. Draw another weird shape as a polygon interior, again using the Sketchpad technique you used in Lab 12. Call the shape X, for ... X-tra weird? Investigate what happens to X as it is transformed in the plane in a specific way.

a. Make a line M1 and a point Q not on M1. Construct a line M2 that intersects M1 and that passes through Q.

b. Mark M1 as a mirror and reflect X in M1 to get X'; then mark M2 as a mirror and reflect X' in M2 to get X''

The rest of #2 is to investigate, as in #1, the effect on a shape of two successive reflections in lines - this time intersecting lines - and to form a conjecture about this combination of reflections. Be sure to include in your investigation:

(i) Dragging a point of X

(ii) Dragging one of the lines

(iii) Rotating M2 about the point of intersection of the two lines - you can just mark the intersection point as a center and use the free-hand rotation tool (one of the Select tool options)

c. State a conjecture or conjectures about the effect on X of two successive reflections in two intersecting lines. Try to be explicit about what transformation is the same as this combination of reflections.

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As a result of the investigations and conjectures in #1 and #2 above, make a statement about reflection as a transformation of points in the plane.

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3. Make 4 copies of W (or of X), print them, and cut them out for use in class tomorrow. They should be fairly small to fit easily on a desk chair, but large enough that you can cut each out and get hold of each.