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:
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.
• 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-AngleAngle 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