Chapter 14: Geometry of Motion and Change Section 14.1: Reflections, Translations, and Rotations

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

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’.

Examples with points

Example with a shape

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.

Example with a shape

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

Example with a shape 180 degree rotation about the point P: not the same as a reflection across the vertical line through P

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.

Why are these 4 transformations important? When applying any of the four transformations of reflection, translation, rotation, or glide reflection: The distance between P and Q is equal to the distance between P’ and Q’. 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.

For practice problems, see Activities 14B and 14C

Section 14.2: Symmetry

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

Examples of Reflectional Symmetry

Examples of Reflectional Symmetry

Rotational Symmetry Def: A shape or design in a plane has rotational symmetry if there is a rotation of the plane of degree >0 and <360 such that the shape occupies the same location after the rotation. It has n-fold rotational symmetry if a 360°/𝑛 rotation moves it to the same location.

Examples of Rotational Symmetry

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.

Examples of Translational Symmetry

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.

What Symmetries exist in the following objects?

Section 14.3: Congruence

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.

Example Ex 1: The hexagons A and B are congruent to each other.

Example 1 cont’d B is a translation of A along the vector v, followed by a reflection across the line L.

See Activity 14 I

Congruence Criteria Side-Side-Side (SSS) Congruence Criterion: Triangles with sides of length 𝑎, 𝑏, and 𝑐 units are all congruent.

Importance of SSS Criterion Triangles are rigid shapes, meaning they are useful for constructing objects that need stable support structures.

Are any side lengths possible for a triangle? Triangle inequality: Assuming 𝑎, 𝑏, and 𝑐 are the side of a triangle with 𝑎≥𝑏≥𝑐, the following inequality must be true: 𝑎<𝑏+𝑐

Congruence Criteria Angle-Side-Angle (ASA) Congruence Criterion: All triangles with a specific side length 𝑎 and angles measuring 𝛼 and 𝛽 degrees at the endpoints of that side are congruent. Need 𝛼+𝛽<180°.

Congruence Criteria Side-Angle-Side (SAS) Congruence Criterion: Triangles with 2 given side lengths 𝑎 and 𝑏 and the angle between those sides being 𝛼 degrees are all congruent.

Other Criteria? Side-Side-Angle, Angle-Angle-Side, and Angle-Angle- Angle are not criteria that force triangles to be congruent.

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.

Section 14.5: Similarity

Ex 1: These two stars are similar.

Ex 1: These two stars are similar.

Ex 1: These two stars are not similar.

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 𝑘 such that the distance between 2 points is 𝑘 times as long on the second object than between the 2 corresponding points on the first object 𝑘 is called the scale factor. All shapes that are congruent are also similar (𝑘=1), but not vice versa.

Ex 1: These two stars are similar with scale factor 𝑘=2.

Note: Scale factors only apply to lengths, and should not be used for areas or volumes.

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?

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?

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?

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.

Section 14.6: Areas, Volumes, and Scaling

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.

Scaling Areas and Volumes For a right triangle or rectangle, scaling the base & height or the length & width by a factor of 𝑘 scales the area by a factor of 𝑘 2 For a rectangular box, scaling 𝑙, 𝑤, & ℎ by a factor of 𝑘 scales the volumes by 𝑘 3