section 12.1 translations and rotationskilmer/36609bn12.pdffollowed by a reﬂection in a line...

c©Kendra Kilmer June 23, 2009

Section 12.1 Translations and Rotations

Any rigid motion that preserves length or distance is anisometry. We look at two types of isome-tries in this section:translations androtations.

Translations

A translation is a motion of a plane that moves every point of the plane in a specified distance in a specifieddirection along a straight line (which can be shown by a slide-arrow or vector).

Example 1: Find the image ofAB under the translation fromX to X ′ pictured on the dot paper below.

Properties of Translations

• A figure and its image are congruent.

• The image of a line is a line parallel to it.

Constructions of Translations

To construct the imageA′ of point A in the direction and magnitude of vector−−→MN, construct a

parallelogramMAA′N so that−→AA′ is in the same direction as

−−→MN

Example 2: Given pointA and vector−−→MN, construct the image,A′, of A.

s

A

M N-

1


Coordinate Representation of Translations

Formulas can be used when the translation is done in the rectangular coordinate system.

Example 3: Let’s look at the following translation in a rectangular coordinate system.

Property of a Translation in a Coordinate System

A translation is a function from the plane to the plane such that to every point(x,y) correspondsthe point(x+a,y+b), wherea andb are real numbers.

Example 4: Find the coordinates of the image of the vertices of quadrilateralABCD under each of the fol-lowing translations:

a) (x,y)→ (x−2,y+4)

b) A translation determined by the slide arrow from(4,−3) to (2,1).

2


Rotation

A rotation is a transformation of the plane determined by rotating the plane about a fixed point, thecenter, by a certain amount in a certain direction. Usually apositive measure is a counterclockwiseturn and a negative measure is a clockwise turn.

Example 5: Find the image of△ABC under the rotation with centerO.

Construction of a Rotation

To construct the image of pointP under a rotation with centerO through a given angleA in thedirection indicated:

• Construct an isosceles triangleBAC with B on one side of the given angle andC on the other side so thatAB=AC=OP.

• Construct△POP′ congruent to△BAC.

Example 6: Construct the image of pointP under the rotation with centerO through the angle and in thedirection given below:

s

O

s

P

-��

A

A rotation of 360◦ about a point will move any point (and figure) onto itself. Such a transformationis anidentity transformation . A rotation of 180◦ about a point is ahalf-turn .

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Slopes of Perpendicular Lines

Theorem: If y1 = m1x+b1 andy2 = m2x+b2 are two distinct non-vertical lines, then

a) m1 = m2 iff the lines are parallel

b) m1m2 =−1 iff the lines are perpendicular.

Example 7: Prove part(b) of the above theorem.

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Example 8: Are the lines 2x−3y = 7 and 3x−2y = 5 parallel, perpendicular, or neither?

Example 9: Find the equation of the line through(−3,2) and perpendicular to the line 3x+ y = 4.

Section 12.1 Homework Problems: 1-8, 11, 13, 18, 19, 22, 33, 34

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Section 12.2 - Reflections and Glide Reflections

Reflections

A reflection is an isometry in which the figure is reflected across areflecting line, creating a mirrorimage. Unlike a translation or rotation, the reflection reverses the orientation of the original figurebut the reflected figure is still congruent to the original figure.

Example 1: Reflect△ABC about the linel.

A

BC

l

Definiton of Reflection

A reflection in a line l is a transformation of a plane that pairs each pointP of the plane with apointP′ in such a way thatl is the perpendicular bisector ofPP′, as long asP is not onl. If P is onl, thenP = P′.

Example 2: Construct the image of pointP under a reflection aboutl.

l

P

6


Example 3: Find the line of reflection of the pointP and its image,P′.

P

P’

Example 4: Construct the image of←→AB under a reflection in linem.

m

B

A

7


Constructing a Reflection on Dot Paper

Example 5: Find the image of each△ABC under a reflection in linel.

a) l

A B

C

b) A

B

C

l

Reflections in a Coordinate System

Example 6: Reflect the pointP across the indicated line and give the coordinates.

a) x-axis

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b) y-axis

c) y = x

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d) y =−x

Example 7: Reflect△KLM across the liney = x−2

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Glide Reflections:

A glide reflection is another basic isometry. It is a transformation consisting of a translationfollowed by a reflection in a line parallel to the slide arrow.

Example 8: Construct the image of△ABC under a glide reflection of slide arrowl.

A

B

C

l

Section 12.2 Homework Problems: 1, 4, 5, 9-11, 13, 14, 35-38

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Section 12.3 - Size Transformations

Definition of Size Transformation

A size transformation from the plane to the plane with centerO and scale factorr (r > 0) isa transformation that assigns to each pointA in the plane a pointA′, such thatO, A, andA′ arecollinear andOA′ = r ·OA and so thatOA is not betweenA andA′.

O

A

BC

A’

B’C’

Example 1: Find the image of△ABC under the size transformation with centerO and scale factor12

A

B

C

O

12


Example 2: Find the image of△ABC under the size transformation with centerO and scale factor 1.5

A

B

C

O

Example 3: Construct the image of pointP under a size transformation with centerO and scale factor12

P

O

13


Example 4: Construct the image of pointP under a size transformation with centerO and scale factor 2

P

O

Example 5: Construct the image of pointP under a size transformation with centerO and scale factor23

P

O

14


Example 6: Construct the image of quadrilateralABCD under a size transformation with centerO and scalefactor 2

3

O

A

B

C

D

Theorem 12-1:

A size transformation with centerO and scale factorr (r > 0) has the following properties:

1. The image of a line segment is a line segment parallel to theoriginal segment andr times as long.

2. The image of an angle is an angle congruent to the original angle.

Definition of Similar Figures:

Two figures are similar if it is possible to transform one ontothe other by a sequence of isometriesfollowed by a size transformation.

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Example 7: Show that△ABC is similar to△A′B′C′ by showing that△A′B′C′ can be found by performing asequence of isometries followed by a size transformation to△ABC.

A

C

A’B B’

C’

Example 8: Show that△ABC is similar to△A′B′C′ by showing that△A′B′C′ can be found by performing asequence of isometries followed by a size transformation to△ABC.

A

A’

B

C

C’

B’

Section 12.3 Homework Problems: 1-4, 6-8, 13, 15, 21, 22

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Section 12.4 - Symmetries

Line Symmetries

A plane region has a linel of symmetry if a reflection of the plane aboutl produces exactly thesame figure.

Example 1: How many lines of symmetry does each object have? Draw the lines of symmetry.

Rotational (Turn) Symmetries

A figure hasrotational symmetry, or turn symmetry , when the traced figure can be rotated lessthan 360◦ about some pointP, theturn center, so that it matches the original figure.

Example 2: Find the pointP and the rotational symmetry for an equilateral triangle.

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Example 3: Determine the amount of turn for the rotational symmetries for each of the figures shown below,if they exist.

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Point Symmetry

Any figure that has 180◦ rotational symmetry is said to havepoint symmetry about the turn center.Any figure with point symmetry is its own image under a half-turn. This makes the center of thehalf-turn the midpoint of a segment connecting a point and its image.

Example 4: Determine whether or not the following figures have point symmetry.

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Symmetries of Three-Dimensional Figures

A three-dimensional figure has aplane of symmetrywhen every point of the figure on one side ofthe plane has a mirror image on the other side of the plane.

Example 5: Determine whether or not each figure has a plane of symmetry.

ss

s

h

r

h

l

b

20


Example 6: Describe what kind of triangle has exactly one line of symmetry and no turn symmetries.

Example 7: Given an arc of a circle, find its center and radius.

Section 12.4 Homework Problems: 2-10, 12, 13b

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Section 12.5 - Tessellations of the Plane

A tessellationof a plane is the filling of the plane with repetitions of figures in such a way that nofigures overlap and there are no gaps.

Example 1: Create a tessellation with a square on the grid below:

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Regular Tessellations

A regular tesselationis a tessellation made up of one type of regular polygon.

Example 2: Try creating tessellations with some basic regular polygons. Which regular polygons tessellatethe plane?

23


.

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Semiregular Tessellations

When more than one type of regular polygon is used and the arrangement of the polygons at eachvertex is the same, the tessellation issemiregular.

Example 3: Create a semiregular tessellation consisting only of squares and equilateral triangles.

Tessellating with Other Shapes

There has been a lot of thought put into determining what shapes (other than regular polygons)tessellate a plane.

Example 4: Does any quadrilateral tessellate a plane?

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Example 5: Let’s see how motion geometry can be used to find other shapes that tessellate a plane.

Section 12.5 Homework Problems: 1-6, 8, 12, 13

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