Download Unit 5.1.1: Defining Transformation Terms

Introduction
Geometric figures can be graphed in the coordinate
plane, as well as manipulated. However, before sliding
and reflecting figures, the definitions of some important
concepts must be discussed.
Each of the manipulations that will be discussed will
move points along a parallel line, a perpendicular line, or
a circular arc. In this lesson, each of these paths and
their components will be introduced.
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5.1.1: Defining Terms
Key Concepts
• A point is not something with dimension; a point is a
“somewhere.” A point is an exact position or location
in a given plane. In the coordinate plane, these
locations are referred to with an ordered pair (x, y),
which tells us where the point is horizontally and
vertically. The symbol A (x, y) is used to represent
point A at the location (x, y).
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5.1.1: Defining Terms
Key Concepts, continued
• A line requires two points to be defined. A line is the
set of points between two reference points and the
infinite number of points that continue beyond those
two points in either direction. A line is infinite, without
beginning or end. This is shown in the diagram below
with the use of arrows. The symbol
is used to
represent line AB.
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5.1.1: Defining Terms
Key Concepts, continued
• You can find the linear distance between two points
on a given line. Distance along a line is written as
d(PQ) where P and Q are points on a line.
• Like a line, a ray is defined by two points; however, a
ray has only one endpoint. The symbol
is used to
represent ray AB.
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5.1.1: Defining Terms
Key Concepts, continued
• Similarly, a line segment is also defined by two
points, but both of those points are endpoints. A line
segment can be measured because it has two
endpoints and finite length. Line segments are used to
form geometric figures. The symbol AB is used to
represent line segment AB.
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5.1.1: Defining Terms
Key Concepts, continued
• An angle is formed where two line segments or rays share
an endpoint, or where a line intersects with another line,
ray, or line segment. The difference in direction of the parts
is called the angle. Angles can be measured in degrees or
radians. The symbol ÐA is used to represent angle A. A
represents the vertex of the angle. Sometimes it is
necessary to use three letters to avoid confusion. In the
diagram below, ÐBAC can be used to represent the same
angle, ÐA. Notice that A is the vertex of the angle and it
will always be listed in between the points on the angle’s
rays.
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5.1.1: Defining Terms
Key Concepts, continued
•
An acute angle measures less than 90° but greater
than 0°. An obtuse angle measures greater than
90° but less than 180°. A right angle measures
exactly 90°.
•
Two relationships between lines that will help us define
transformations are parallel and perpendicular.
Parallel lines are two lines that have unique points
and never cross. If parallel lines share one point, then
they will share every point; in other words, a line is
parallel to itself.
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5.1.1: Defining Terms
Key Concepts, continued
•
Perpendicular lines meet at a right angle (90°),
creating four right angles.
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5.1.1: Defining Terms
Key Concepts, continued
•
A circle is the set of points on a plane at a certain
distance, or radius, from a single point, the center.
Notice that a radius is a line segment. Therefore, if
we draw any two radii of a circle, we create an angle
where the two radii share a common endpoint, the
center of the circle.
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5.1.1: Defining Terms
Key Concepts, continued
•
Creating an angle inside a circle allows us to define a
circular arc, the set of points along the circle
between the endpoints of the radii that are not
shared. The arc length, or distance along a circular
arc, is dependent on the length of the radius and the
angle that creates the arc—the greater the radius or
angle, the longer the arc.
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5.1.1: Defining Terms
Common Errors/Misconceptions
• mislabeling angles or not including enough points to
specify an angle
• misusing terms and notations
• finding the length of incorrect arcs
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5.1.1: Defining Terms
Guided Practice
Example 4
Given the following:
AC @ BD
WY < XZ
AB ^ AC
WX ^ WY
AB ^ BD
WX ^ XZ
Are AB and CD parallel? Are WX and YZ parallel? Explain.
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5.1.1: Defining Terms
Guided Practice: Example 4, continued
1. AC and BD intersect AB at the same
angle and AC @ BD.
will never cross
to CD.
. Therefore, AB is parallel
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5.1.1: Defining Terms
Guided Practice: Example 4, continued
2. WY and XZ intersect WX at the same
angle, but WY < XZ .
As you move from Z to Y on
and will eventually intersect,
not parallel to YZ .
, you move closer to,
. Therefore, WX is
✔
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5.1.1: Defining Terms
Guided Practice: Example 4, continued
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5.1.1: Defining Terms
Guided Practice
Example 5
Refer to the figures below. Given AB @ BC , is the set of
points with center B a circle? Given XY > YZ , is the set
of points with center Y a circle?
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5.1.1: Defining Terms
Guided Practice: Example 5, continued
1. The set of points with center B is a circle
because all points are equidistant from
the center, B.
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5.1.1: Defining Terms
Guided Practice: Example 5, continued
2. The set of points with center Y is not a
circle because the points vary in distance
from the center, Y.
✔
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5.1.1: Defining Terms
Guided Practice: Example 5, continued
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5.1.1: Defining Terms