c Kendra Kilmer June 23, 2009 Section 12.1 Translations and Rotations Any rigid motion that preserves length or distance is an isometry. We look at two types of isome- tries in this section: translations and rotations. Translations A translation is a motion of a plane that moves every point of the plane in a specified distance in a specified direction along a straight line (which can be shown by a slide-arrow or vector). Example 1: Find the image of AB under the translation from X 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 image A ′ of point A in the direction and magnitude of vector −−→ MN , construct a parallelogram MAA ′ N so that −→ AA ′ is in the same direction as −−→ MN Example 2: Given point A and vector −−→ MN, construct the image, A ′ , of A. A M N ✲ 1
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Section 12.1 Translations and Rotationskilmer/36609bn12.pdffollowed by a reflection in a line parallel to the slide arrow. Example 8: Construct the image of ABC under a glide reflection
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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.
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
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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 .
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.
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 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
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 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.
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.
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.