Download - Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations
![Page 1: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/1.jpg)
Chapter 14: Geometry of Motion
and Change
Section 14.1: Reflections, Translations, and Rotations
![Page 2: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/2.jpg)
Transformations
• Def: A transformation of a plane is an action that changes or transforms the plane.
•We will look at transformations that result in the same plane but with points in it rearranged in some way.
• 3 Major types: reflections, translations, and rotations
![Page 3: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/3.jpg)
Reflections
• Def: A reflection (or flip) of a plane across a chosen line, called the line of reflection ℓ, results in the following for each point P:
P is moved across ℓ along a line through P that is perpendicular to ℓ so that P remains the same distance from ℓ but on the other side of the line.•We call the resulting point P’.
![Page 4: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/4.jpg)
Exampleswith points
![Page 5: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/5.jpg)
Example with a shape
![Page 6: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/6.jpg)
Translations• Def: A translation (slide) is the result of moving each point in
the plane a given distance in a given direction, as described by the translation vector v.
![Page 7: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/7.jpg)
Example with a shape
![Page 8: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/8.jpg)
Rotations
• Def: A rotation (turn) results in each point in the plan rotating about a fixed point by a fixed angle.
• Ex: A 90 degree rotation (clockwise) about the point A
![Page 9: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/9.jpg)
Example with a shape
• 180 degree rotation about the point P: not the same as a reflection across the vertical line through P
![Page 10: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/10.jpg)
Glide Reflection
• Def: A glide reflection is the result of combining a reflection with a translation in the direction of the line of the reflection.
![Page 11: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/11.jpg)
Why are these 4 transformations important?•When applying any of the four transformations of reflection,
translation, rotation, or glide reflection: 1. The distance between P and Q is equal to the distance
between P’ and Q’.2. The angle PQR is the same as the angle P’Q’R’
• Any transformation that preserves these 2 facts is one of the four that we defined.
![Page 12: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/12.jpg)
• For practice problems, see Activities 14B and 14C
![Page 13: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/13.jpg)
Section 14.2: Symmetry
![Page 14: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/14.jpg)
Reflectional Symmetry
• Def: A shape or design in a plane has reflectional symmetry if the shape occupies the exact same location after reflecting across a line, called the line of reflection.
• Alternatively, the two sides of the shape match when folded along the line of symmetry
![Page 15: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/15.jpg)
Examples of Reflectional Symmetry
![Page 16: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/16.jpg)
Examples of Reflectional Symmetry
![Page 17: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/17.jpg)
Rotational Symmetry
• Def: A shape or design in a plane has rotational symmetry if there is a rotation of the plane of degree and such that the shape occupies the same location after the rotation.
• It has n-fold rotational symmetry if a rotation moves it to the same location.
![Page 18: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/18.jpg)
Examples of Rotational Symmetry
![Page 19: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/19.jpg)
Translational Symmetry
• Def: A design or pattern in a plane has translational symmetry if there is a translation of the plane such that the pattern as a whole occupies the same place after applying the translation.
• The pattern can not simply be a shape because it must take up an entire line or the entire plane.
![Page 20: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/20.jpg)
Examples of Translational Symmetry
![Page 21: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/21.jpg)
Glide Reflection Symmetry
• Def: A design or pattern has glide reflection symmetry if there is a reflection followed by a translation after which the design occupies the same location.
![Page 22: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/22.jpg)
What Symmetries exist in the following objects?
![Page 23: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/23.jpg)
Section 14.3: Congruence
![Page 24: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/24.jpg)
Definition of Congruence
•Def: Two shapes or designs are congruent if there is a rotation, reflection, translation, or combination of these 3 that transforms one shape into the other.
![Page 25: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/25.jpg)
Example• Ex 1: The hexagons A and B are congruent to each other.
![Page 26: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/26.jpg)
Example 1 cont’d
B is a translation of A along the vector v, followed by a reflectionacross the line L.
![Page 27: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/27.jpg)
See Activity 14 I
![Page 28: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/28.jpg)
Congruence Criteria
Side-Side-Side (SSS) Congruence Criterion:Triangles with sides of length , and units are all congruent.
![Page 29: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/29.jpg)
Importance of SSS Criterion
Triangles are rigid shapes, meaning they are useful for constructing objects that need stable support structures.
![Page 30: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/30.jpg)
Are any side lengths possible for a triangle?Triangle inequality: Assuming , , and are the side of a triangle with , the following inequality must be true:
![Page 31: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/31.jpg)
Congruence Criteria
Angle-Side-Angle (ASA) Congruence Criterion:All triangles with a specific side length and angles measuring and degrees at the endpoints of that side are congruent.
Need .
![Page 32: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/32.jpg)
Congruence Criteria
Side-Angle-Side (SAS) Congruence Criterion:Triangles with 2 given side lengths and and the angle between those sides being degrees are all congruent.
![Page 33: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/33.jpg)
Other Criteria?
Side-Side-Angle, Angle-Angle-Side, and Angle-Angle-Angle are not criteria that force triangles to be congruent.
![Page 34: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/34.jpg)
Application to facts about parallelograms•Recall: A parallelogram is a quadrilateral with opposite
sides being parallel.
•Alternative definition: a quadrilateral with opposite sides being the same length.
• See Activity 14K for why these are equivalent.
![Page 35: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/35.jpg)
Section 14.5: Similarity
![Page 36: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/36.jpg)
Ex 1: These two stars are similar.
![Page 37: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/37.jpg)
Ex 1: These two stars are similar.
![Page 38: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/38.jpg)
Ex 1: These two stars are not similar.
![Page 39: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/39.jpg)
Definition of Similarity
• Def: Two shapes or objects (in a plane or space) are similar if every point on one object corresponds to a point on the other object and there is a positive number such that the distance between 2 points is times as long on the second object than between the 2 corresponding points on the first object• is called the scale factor.• All shapes that are congruent are also similar (), but not vice versa.
![Page 40: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/40.jpg)
Ex 1: These two stars are similar with scale factor
![Page 41: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/41.jpg)
Note: Scale factors only apply to lengths, and should not be used for areas or volumes.
![Page 42: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/42.jpg)
3 Methods for solving similar objects problems• Scale Factor Method: find scale factor and multiply/ divide to solve
• Ex 2: The Khalifa Tower in Dubai is the tallest building in the world at about 2700 feet tall. If a scale model of the building is 9 feet tall and 1 foot 10 inches wide at the base, what is the width of the base of the actual building?
![Page 43: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/43.jpg)
3 Methods for solving similar objects problems• Internal Factor Method: use internal comparisons within each shape
• Ex 3: If a model airplane measures 8 inches from the front to the tail (length) and 4 inches for the wingspan, what is the wingspan of an actual plane that is 24 feet 6 inches long?
![Page 44: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/44.jpg)
3 Methods for solving similar objects problems• Proportion Method: solve using proportional equations
• Ex 3 again: If a model airplane measures 8 inches from the front to the tail (length) and 4 inches for the wingspan, what is the wingspan of an actual plane that is 24 feet 6 inches long?
![Page 45: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/45.jpg)
Triangle Similarity Criteria
• Angle-Angle-Angle Similarity Criterion for Triangle Similarity: Two triangles are similar exactly when they have the same size angles.
• There are many special cases of when this similarity occurs.
![Page 46: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/46.jpg)
Section 14.6: Areas, Volumes, and
Scaling
![Page 47: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/47.jpg)
Example Problem
• The following figures show a cylinder and the same cylinder scaled by a factor of 2. Their volume is scaled by a factor that is larger than 2.
![Page 48: Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations](https://reader033.vdocuments.mx/reader033/viewer/2022061616/56649ddb5503460f94ad282d/html5/thumbnails/48.jpg)
Scaling Areas and Volumes
• For a right triangle or rectangle, scaling the base & height or the length & width by a factor of scales the area by a factor of
• For a rectangular box, scaling by a factor of scales the volumes by