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Prove synthetically that every rotation A, is an isometry.
4.2.8 Theorem. Any isometry of the Euclidean plane can be written as one of the following transformations: reflection, rotation, translation or glide reflection.

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Prove that successive reflections in parallel mirror lines produce a translation in a direction perpendicular to the mirrors through a distance equal to twice the distance between the mirrors.

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Suppose you wish to join the two towns A (1,5) and B(8,2) via a pipeline. A pumping station is to be placed along a straight riverbank (the x-axis). Determine the location of a pumping station, P(x,0), that minimizes the amount of pipe used? Solve this

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Buried Treasure. Among his great-grandfathers papers, José found a parchment describing the location of a hidden treasure. The treasure was buried by a band of pirates on a deserted island which contained an oak tree, a pine tree, and gallows where the pirates hanged traitors. The map looked like the accompanying figure and gave the following directions. Count the steps from the gallows to the oak tree. At the oak, turn 90° to the right. Take the same number of steps and then put a spike in the ground. Next, return to the gallows and walk to the pine tree, counting the number of steps. At the pine tree, turn 90° to the left, take the same number of steps, and then put another spike in the ground. The treasure is buried halfway between the spikes. José found the island and the trees but could not find the gallows or the spikes, which had long since rotted. José dug all over the island, but because the island was large, he gave up. Devise a plan to help José find the treasure.

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Recall the two-regular tilings of order 2 produced with squares and triangles. Classify each as a wallpaper design.

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Classify the following wallpaper design. Is there any relation to the checkerboard tiling?

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What type of wallpaper design is Eschers version of Devils and Angels for Euclidean geometry?

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On sketchpad use custom transformations to create a wallpaper design other than a p4g.
*Different Symmetry (Glide Reflection)

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Let ABCD be a quadrilateral. In the figure below E, F,G, and H are the midpoints of the sides. Prove that EFGH is a parallelogram. Hint: Similar triangles.

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Eschers lizard graphic is shown below. Mark all the points in the picture about which there are rotations by 180˚. What do you notice about these points? Exhibit a parallelogram and three points about which successive rotations through 180˚ would produce Escher’s design. What is the wallpaper classification for the lizard design?