College Algebra Exam Review 2

College Algebra Exam Review 2 - 12 1. ALGEBRAIC THEMES map...

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12 1. ALGEBRAIC THEMES map vertices to vertices!) Now there is exactly one point in R that is equidistant from the four vertices; this is the centroid of the ﬁgure, which is the intersection of the two diagonals of R . Denote the centroid C . What is ±.C/ ? Since ± is an isometry and maps the set of vertices to itself, ±.C/ is still equidistant from the four vertices, so ±.C/ D C . We can assume without loss of generality that the ﬁgure is located with its centroid at 0 , the origin of coordinates. It follows from the results quoted in the previous paragraph that ± extends to a linear isometry of R 3 . The same argument and the same conclusion are valid for many other geometric ﬁgures (for example, polygons in the plane, or polyhedra in space). For such ﬁgures, there is (at least) one point that is mapped to itself by every symmetry of the ﬁgure. If we place such a point at the origin of coordinates, then every symmetry of the ﬁgure extends to a linear isometry of R 3 . Let’s summarize with a proposition:
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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