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Which one of these is not an isometry? A. a translation B. a 30 degrees rotation about...

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t4trendesetter | Honors

Posted May 15, 2013 at 8:57 PM via web

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Which one of these is not an isometry?

A. a translation B. a 30 degrees rotation about the orgin followed by a flection in the x-axis C. a dilation D. a reflection in the line y = -x followed by a 160 degrees rotation about the point (1, -2)

 

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crmhaske | College Teacher | (Level 3) Associate Educator

Posted May 16, 2013 at 12:22 AM (Answer #1)

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An isometry is when a new spatial map is created from an old spatial map, while still preserving the distance between elements in the map.

Therefore

A translation is an isometry because all elements in the map are moved an equal amount.

A rotation is an isometry because all elements in the map are rotated an equal amount, and a flexion is an isometry because all elements in the map are bent around the point of flexion equally.

Consequently,

A, B, and D are all isometries as they involve translation, rotation, and flexion, all of which preserve the distance between elements in the map.

This leaves us with C. dilation.  By the process of elimination it must not be an isometry.  This is true because a dilation involves expanding the map outwards from the origin of the dilation.  In the process of doing so, the distance between the elements in the map increases, and therefore, a dilation cannot be an isometry

 

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crmhaske | College Teacher | (Level 3) Associate Educator

Posted May 16, 2013 at 12:23 AM (Answer #2)

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I missed reflection.  A reflection is an isometry because it involves a mirror image of the elements in the map, and so therefore does not alter the distance between those elements.

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