Section 12.1 – Translations and Rotations
Any rigid motion that preserves length or distance is an isometry (meaning “equal
measure”). In this section, we will investigate two types of isometries: translations and
rotations.
Translations
• A translation is a motion of a plane that moves every point of the plane a
specified distance in a specified direction along a straight line.
• Properties of Translations
o A figure and its image are congruent.
o The image of a line is a line parallel to it.
Example: Find the image of AB under the translation from X to X′ pictured on the dot
paper below.
B
A
X
X′
Constructions of Translations
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Example: Construct the image A′ of point A in the direction and magnitude of vector MN .
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Coordinate Representation of Translations
You can use formulas when the translation occurs in the 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)
corresponds the point (x + a, y + b), where a and b are real numbers.
Example: Find the coordinates of the image of the vertices of quadrilateral ABCD in the
figure below under the translations given. Draw the image in each case.
•
(x, y) → (x + 2, y – 3)
•
a translation determined by the slide arrow from A(1, 2) to A′ (3, −1)
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Rotations
• A rotation is a transformation of the plane determined by rotating the plane about
a fixed point, the center, by a certain amount in a certain direction.
• Usually, a positive measure is a counterclockwise turn and a negative measure is a
clockwise turn.
• A rotation of 360° about a point will move any point (and figure) onto itself. Such
a transformation is an identity transformation. A rotation of 180° is a half-turn.
Example: Find the image of ABC under the rotation with center O.
A
B
C
Construction of a Rotation
Example: Construct the image of point P under a rotation with center O through the angle
and in the direction given in the figure below.
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Slopes of Perpendicular Lines
Two lines, neither of which is vertical, are perpendicular iff their slopes m1 and m2 satisfy
the condition m1m2 = -1. Every vertical line has an undefined slope and is perpendicular
to a line with slope 0.
Theorem: If y1 = m1x+b1 and y2 = 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: Prove part b of the above theorem.
Example: Are the lines 2x – 3y = 7 and 3x – 2y = 5 parallel, perpendicular, or neither?
Example: Find the equation of the line through (–3, 2) and perpendicular to the line 3x + y = 4.
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Section 12.2 – Reflections and Glide Reflections
Reflections
• A reflection is an isometry in which a figure is reflected across a reflecting line,
creating a mirror image.
o Unlike a translation or rotation, the reflection reverses the orientation of
the original figure.
o The reflected figure is congruent to the original figure.
Example: Reflect ABC about the line l.
Definition of Reflection
A reflection in a line l is a transformation of a plane that pairs each point P of the plane
with a point P′ in such a way that l is the perpendicular bisector of PP! , as long as P is
not on l. If P is on l, then P = P′.
Construction of a Reflection
Example: Construct the image of point P under a reflection about l.
P
l
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Example: Find the line of reflection.
Reflections in a Coordinate System
Example: Find the image of each ABC under a reflection in line l.
a.
b.
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Example: Reflect the point P across the given line and give the coordinates.
a. y-axis
b. x-axis
c. y = x
d. y = −x
Example: Reflect ΔKLM across the line y = x – 2.
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Glide Reflections
A glide reflection is another basic isometry. A glide reflection is a transformation
consisting of a translation followed by a reflection in a line parallel to the slide arrow.
Example: Construct the image of ABC under a glide reflection of slide arrow l.
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Section 12.3 – Size Transformations
Definition of Size Transformation
A size transformation from the plane to the plane with center O and scale factor r (r > 0)
is a transformation that assigns to each point in the plane a point A′ such that O, A, and A′
are collinear and OA′ = r⋅OA and so that O is not between A and A′.
Theorem 12-1
A size transformation with center O and scale factor r (r > 0) has the following
properties:
1. The image of a line segment is a line segment parallel to the original segment
and r times as long.
2. The image of an angle is an angle congruent to the original angle.
Example: In each figure below, show that ABC is the image of DEF under a size
transformation. Identify the center of the size transformation and the scale factor.
a.
b.
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Example: Find the image of ABC under a size transformation with center O and scale factor
1
2
Example: Find the image of ABC under a size transformation with center O and scale
factor 1.5.
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.
Example: Construct the image of quadrilateral ABCD under a size transformation with
center O and scale factor 23 .
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Definition of Similar Figures
Two figures are similar if it is possible to transform one onto the other by a sequence of
isometries (translation, rotation, reflection) followed by a size transformation (dilation).
Example: Show that ABC is similar to A′B′C′ by showing that A′B′C′ can be found
by performing a sequence of isometries followed by a size transformation to ABC.
a.
b.
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Section 12.4 – Symmetries
Line Symmetries
A plane region has a line of symmetry l if it is its own image under a reflection in l.
Example: How many lines of symmetry does each object have? Draw the lines of symmetry.
H
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M
Rotational (Turn) Symmetries
A figure has rotational symmetry, or turn symmetry, when the traced figure can be rotated
less than 360° about some point P, the turn center, so that it matches the original figure.
Example: Find the point P and the rotational symmetry for an equilateral triangle.
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Example: Determine the amount of the turn for the rotational symmetries of each of the
figures below.
Z
Point Symmetry
Any figure that has 180° rotational symmetry is said to have point symmetry about the
turn center. Any figure with point symmetry is its own image under a half-turn. This
makes the center of the half-turn the midpoint of a segment connecting a point and its
image.
Example: What objects from previous examples have point symmetry?
Example: What kind of triangle has exactly one line of symmetry and no turn symmetries?
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Symmetries of Three-Dimensional Figures
A three-dimensional figure has a plane of symmetry when every point of the figure on
one side of the plane has a mirror image on the other side of the plane. Three-dimensional
figures with rotational symmetry rotate about an axis of rotation.
Example: Determine whether or not each figure has a plane of symmetry.
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Section 12.5 – Tessellations of the Plane
A tessellation of a plane (or space) is the filling of the plane (or space) with repetitions of
figures in such a way that no figures overlap and there are no gaps.
John Locke
M.C. Escher
Example: Create a tessellation with a square on the grid below:
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Regular Tesselations
• A regular tessellation is a tessellation made up of one type of regular polygon.
• If a regular polygon tessellates the plane, the sum of the congruent angles of the
polygons around every vertex must be 360°.
Example: Try creating tessellations with some basic regular polygons. Which regular
polygons tessellate the plane?
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Semiregular Tessellations
When more than one type of regular polygon is used and the arrangement of the polygons
at each vertex is the same the tessellation is semiregular.
Example: Create a semiregular tessellation consisting only of squares and equilateral triangles.
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Tessellations with Other Shapes
Example: Does every quadrilateral tessellate a plane?
Example: Use motion geometry to find other shapes that tessellate a plane.
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