Lab 10: Identifying Isometries

This lab is devoted to exploring more about composition of isometries and identifying isometries.

For some ideas about identification using orbits, refer back to Lab 8.

Part A. Midpoints and Midlines (i.e., perpendicular bisectors)

Orientation preserving and reversing isometries each have distinguishing traits.

• Download this Sketchpad file and follow the directions to explore these properties.

Part B. Mystery Isometries

Test your isometry toolkit by constructing the geometric defining data of these 3 isometries.

• Download the GSP file.

Part C. Triple Line Reflections

In a new sketch, draw 3 points A, B, C and the 3 lines a = BC, b = CA, c = AB.

• Let G be the product of line reflections R_c R_b R_a .
• Investigate G and construct the geometric defining data for G.
• If H is the product of line reflections R_a R_b R_c , how is H related to G?

Part D. Compositiion of rotations

Start a new sketch with points A and B. Draw angles EFG and JKL.

Draw a point P and then rotate P by angle EFG with center A to get P' and then rotate P' by angle JKL with center B to get P''.

The composition of the two rotations is a rotation with a center C(except in a special case).

• Construct center C by the "kite" method, not using points P, P', P''. In other words, construct lines so that each rotation is the product of two line reflections and so that the intersection of two of the lines is the center C.

Part E. Square root of -1

Let O be a point and let transformation M be the dilation with center O and ratio -1 (this is also a point symmetry).

Construct a tranformation N so that NN = M. Is there more than one?