Download Algebra 2nd Semester Final Study Guide

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Isometry- does not change shape or size of figure, only changes
location or position
Dilation
Types of transformations:
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Reflection
Rotation
Translation
Dilation
CONSTRUCTING TRANSLATIONS NOTES
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A dilation is not an isometry.
A transformation is written PREIMAGE  IMAGE
The preimage is the original figure. The image is the translated
figure, and is written with primes.
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(x,y)  (ax, by)
Each is multiplied by a scale
factor. Fractional scale
factor/dividing means getting
smaller.
Rotations are always counterclockwise unless specified
otherwise
A composition of transformations is one transformation
followed by others.
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REFLECTIONS NOTES
Reflection across y axis
(x,y)  (-x,y)
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Reflection across x axis
(x,y)  (x, -y)
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Rotation (90 degree)
(x,y)  (y, -x)
Rotation (180 degree)
(x,y)  (-x, -y)
Rotation (270 degree)
(x,y)  (-y, x)
Translation
(x, y)  (x +a, y + b)
A and B are negative/
subtraction for left or down
and positive for right or up.
A vector has distance and direction
When using a vector to translate a figure, whatever part of
the figure is at the tail of the vector will be moved to the tip
of the vector.
Vectors MUST be written with brackets. They’re not a point,
they show change.
Example: <1,2> Shows a movement of one right and two up.
Flip the signs of the vector to go from the image back to the
preimage
Reflections can be used to give you the shortest distance
between two points.
Think of what values are staying the same- x or y- to
determine what line you are reflecting over.
Theorem
The composition of two reflections across two parallel lines is
equivalent to a translation in that same direction.
- The translation vector is perpendicular to the two lines
- The length of the translation vector is twice that of the
distance between the two lines
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TRANSLATIONS NOTES
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Theorem
-The composition of two reflections across intersecting lines is
equivalent to a rotation.
- The center of rotation is the intersection
- The angle of rotation is twice the angle of intersection
Glide translation
-A glide translation is a translation followed by a reflection.
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Frieze pattern
-This is a zig zag pattern.
Tessellation
-A tessellation has angles that add up to 360 degrees in each
place that they meet.
LINE SYMMETRY is when a figure can be reflected over a
line of symmetry to coincide with itself.
ROTATIONAL SYMMETRY is when a figure can be rotated
about a point by an angle less than 360 and greater than 0
degrees, so that it coincides with itself
The ANGLE OF ROTATIONAL SYMMETRY is the smallest
angle you can rotate a figure by for it to coincide with itself
The ORDER is the number of times a figure can be rotated
to coincide with itself. ORDER is 360/angle of rotational
symmetry
PLANE SYMMETRY is when you can divide a 3D figure with a
plane, and both portions of it can be reflected across that
plane to coincide.
SYMMETRY ABOUT AXIS is when a figure can be rotated
about an axis and coincide with itself at some point less
than a 360 degree rotation.
TRANSLATIONAL SYMMETRY is when a pattern can be
translated along a vector and one part coincides with
another.
PERPENDICULAR LINES NOTES
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Two lines that are perpendicular meet at a 90 degree angle
Their slopes cancel out to equal one. For example
m= 3/4 and m= -4/3 are perpendicular
MIDPOINT
SYMMETRY NOTES
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A figure has symmetry if there is a transformation where
the image coincides with the preimage.
A LINE OF SYMMETRY divides a figure into two congruent
halves.
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middle point
the midpoint formula is used to find the coordinates of the
midpoint by averaging the x and y of the two points. This is
written as:
Midpoint Formula
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DISTANCE
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You can find distance by plugging the rise and run into the
Pythagorean theorem as though they are the sides of a
triangle.
You can also use the distance formula, which is derived
from the Pythagorean theorem.
Distance Formula
PARTITIONING LINE SEGMENTS
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To partition a line at a ratio, you need to divide the line into
parts by that ratio.
Example: If there is a 1:3 ratio, then one part will be ¼ the
size of the original line and another part will be ¾ the size of
the original line.
There is a formula to represent this.
Partitioning Segments Formula
At a ratio of A:B, the formula is…
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PROOFS NOTES
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ANGLES
The set of all points between the sides of the angle are
referred to as the INTERIOR of the angle
The EXTERIOR of an angle is the set of all points outside the
angle
An angle can be named by its VERTEX (middle point) or by
any combination of points with the vertex in the middle of
the name.
Example: Angle R, Angle PRT, Angle TRP
Angles can also be numbered
ACUTE angles measure greater than zero and less than 90
degrees.
RIGHT angles measure 90 degrees.
OBTUSE angles measure greater than 90 but less than 180.
STRAIGHT angles measure 180 degrees
The middle of an angle is called the VERTEX
All proofs have a statement and a reason for each step.
Some reasons can include, definitions, postulates,
theorems, properties, “given”, and “simplify”.
All items in a two column proof need to be numbered to
make a list.
Paragraph proofs are considered informal (Two column
proofs are formal)
It is acceptable to use abbreviations during a proof
A proof uses DEDUCTIVE REASONING to create logical steps
Corollary
-The immediate consequence of a result already proven. These
are things you pick up in the process of writing a proof that are a
result of something else you did.
Postulate
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-This is a statement that is taken to be true and does not need to
be proven.
Theorem
-A theorem is a proven statement.
CONDITIONAL STATEMENTS NOTES
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A conditional statement is written as PQ , which is said,
“If P, then Q”
The CONVERSE is written Q P and is the reversed order of
the statement.
The INVERSE is written ~ P  ~Q and is composed of the
negatives of both the hypothesis and conclusion.
Truth value Truth value Truth value
of P
of Q
of statement
T
F
F
T
T
T
F
T
T
F
F
T
A statement is only false if the conclusion is false and the
hypothesis is not.
BICONDITIONAL STATEMENTS are true both forwards and
backwards
This means their converse is also true.