Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 9.5 Transformational Geometry, Symmetry, and Tessellations.

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 9.5

Transformational Geometry, Symmetry, and Tessellations


What You Will Learn

Rigid Motion or TransformationReflectionsTranslationsRotationsGlide ReflectionsSymmetryTessellations

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Definitions

The act of moving a geometric figure from some starting position to some ending position without alteringits shape or size iscalled a rigid motion(or transformation).

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ReflectionA reflection is a rigid motion that moves a geometric figure to a new position such that the figure in the new position is a mirror image of the figure in the starting position. In two dimensions, the figure and its mirror image are equidistant from a line called the reflection line or the axis of reflection.

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Example 1: Reflection of a TriangleConstruct the reflection of triangle ABC about reflection line l.

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SolutionReflection of ABC is A´BĆ´Point A: 2 units to right of l.So A´ is 2 units to left of l.Point B: 6 units to right of l.So B´ is 6 units to left of l.Point C: 6 units to right of l.So C´ is 6 units to left of l.

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Example 1: Reflection of a Triangle


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Example 1: Reflection of a Triangle


Translation

A translation (or glide) is a rigid motion that moves a geometric figure by sliding it along a straight line segment in the plane. The direction and length of the line segment completely determine the translation.

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Translation

A concise way to indicate the direction and the distance that a figure is moved during a translation is with a translation vector.

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Example 2: A Translated SquareGiven square ABCD and translation vector v, construct the translated square A´BĆ´D´.

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Example 2: A Translated SquareSolutionTranslation ofABCD is A´BĆ´DÍt will have samesize and shapeThe translationvector points 6units downwardand 3 units right

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Example 2: A Translated Square

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Rotation

A rotation is a rigid motion performed by rotating a geometric figure in the plane about a specific point, called the rotation point or the center of rotation. The angle through which the object is rotated is called the angle of rotation.

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Example 5: A Rotation Point Inside a PolygonGiven polygon ABCDEFGH and rotation point P, construct polygons that result fromrotations through

a) 90º b) 180º.

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Example 5: A Rotation Point Inside a PolygonSolutiona) Rotate it 90ºcounterclockwiseHorizontal linesegments will bevertical line segmentsVertical line segments will be horizontal line segments

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Example 5: A Rotation Point Inside a PolygonSolution

GH

G His 1 unit above P

is 1 unit left of P GF

G Fis 4 units left of Pis 4 units below P

DE

D E

is 1 unit down and 2 units left of Pis 2 units down and 1 unit right of P

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Example 5: A Rotation Point Inside a Polygon

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Example 5: A Rotation Point Inside a PolygonSolutionb) Rotate it 180ºcounterclockwiseHorizontal linesegments will still behorizontal line segmentsVertical line segments will still be vertical line segments

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Example 5: A Rotation Point Inside a PolygonSolution

GH

G His 1 unit above Pis 1 unit below P

GF

G Fis 4 units left of Pis 4 units right of P

DE

D E

is 1 unit down and 2 units left of Pis 1 unit up and 2 units right of P

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Glide Reflection

A glide reflection is a rigid motion formed by performing a translation (or glide) followed by a reflection.

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Symmetry

A symmetry of a geometric figure is a rigid motion that moves a figure back onto itself. That is, the beginning position and ending position of the figure must be identical.

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Reflective SymmetryIf we use rigid motion reflection about line l, and the ending position is identical to the starting position, although vertex labels are different, we say that the polygon has reflective symmetry about line l. We refer to line l as a line of symmetry.

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For Example

Reflection about line l

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Rotational SymmetryIf we rotate a polygon about a point P, and the ending position is identical to the starting position, although vertex labels are different, we say that the polygon has rotational symmetry about line l. We refer to point P as a point of symmetry.

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For Example

Rotation of 90º about point P

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Tessellations

A tessellation (or tiling) is a pattern consisting of the repeated use of the same geometric figures to entirely cover a plane, leaving no gaps. The geometric figures used are called the tessellating shapes of the tessellation.

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For Example

The simplest tessellations use one single regular polygon.

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For Example

The simplest tessellations use one single regular polygon.

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For Example

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Creating a Tessellation with a SquareBegin with a 2” square, draw a line.

Cut and rotate.

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Draw another line.

Cut and rotate.

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Creating a Tessellation with a Square


Trace the shape on blank piece of paper.Move the shape, lining it up with the drawn figure and trace.Repeat until page is completely covered.

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Creating a Tessellation with a Square

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 9.5 Transformational Geometry, Symmetry, and Tessellations.

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