Download Guided Practice: continued

Chapter 5
Intro to Congruence
Transformations and Coordinate
Proofs
Objectives:


Identify the three types of congruence
transformations.
Use the Cartesian plane to demonstrate
Geometric properties
Introduction

Triangles are typically thought of as simplistic
shapes constructed of three angles and three
segments. As we continue to explore this
shape, we discover there are many more
properties and qualities than we may have
first imagined. Each property and quality,
such as the mid-segment of a
triangle, acts as a tool for solving
problems.
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Key Concepts
•
•
The midpoint is the point on a line segment
that divides the segment into two equal parts.
A midsegment of a triangle is a line segment
that joins the midpoints of two sides of a
triangle.
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Key Concepts, continued
•
•
•
In the diagram below, the midpoint of AB
is X.
The midpoint of BC is Y.
A midsegment of
is XY .
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Key Concepts, continued
•
The midsegment of a triangle is parallel
to the third side of the triangle and is
half as long as the third side. This is
known as the Triangle Midsegment
Theorem.
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
Key Concepts, continued
•
Every triangle has three midsegments.
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

Guided Practice: Example 1
Find the lengths of BC and YZ and the
measure of ∠AXZ.
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Transformations


A transformation is an operation that maps an
original geometric figure (the preimage) onto a
new figure (the image).
Transformations are noted with an arrow.


Example: ABC  XYZ indicates that A is mapped to X,
B is mapped to Y, and C is mapped to Z.
If the preimage and image are congruent
figures, then the transformation is called a
congruence (or rigid) transformation, or an
isometry (iso – “same”)
Congruence Transformations

There are 3 types of congruence transformations. (p. 294)
 A reflection (or flip) is a transformation over a line
called the line of reflection. Each point of the preimage
and its image are the same distance away from the line
of reflection.
 A translation (or slide) is a transformation that moves
all points of the preimage the same distance in the
same direction.
 A rotation (or turn) is a transformation around a fixed
point called the center of rotation, through a specific
angle and in a specific direction. Each point of the
preimage and its image are the same distance from the
center.
Coordinate Proofs


The invention of the Cartesian (or coordinate)
plane created connections between Algebra
and Geometry. This allowed new discoveries
and new techniques for examining geometric
figures.
A coordinate proof uses figures in the
coordinate plane and algebra to prove
geometric concepts.
Coordinate Proofs

To perform a coordinate proof, you must first
position a figure (e.g. a triangle) on the
coordinate plane:




Use the origin as a vertex or center of the triangle.
Place at least one side of the figure on an axis.
Keep the figure within the first quadrant if
possible.
Use coordinates that make computations as simple
as possible.
Observations about
midsegments


Draw the midsegment
connecting sides KL and JL.
How can we do this?
Use the midpoint formula.
K (4, 5)
4
J (-2, 3)
2
N
M
5


Can we make any observations
about the relationship between
the midsegment MN and the
side JK?
(Hint: check their slopes and
lengths.)
L (6, -1)
-2
-4
Midsegment Theorem

The segment connecting
the midpoints of two sides
of a triangle is parallel to
the third side and is half
as long.

C
D
E
DE ║ AB, and
DE = ½ AB
A
B
Using the Midsegment
Theorem



UW and VW are midsegments of
∆RST. Find UW and RT.
UW = ½(RS) = ½ (12) = 6
RT = 2(VW) = 2(8) = 16
R
U


What would RU and UT be?
RV and VS?
12
V
8
T
W
S
Connect all midpoints





What would happen if you drew all
three midsegments?
How do the lengths of the
midsegments compare to the other
lengths?
So what have we formed?
Four congruent triangles (SSS).
How do they relate to the original
triangle?
They are all similar to the original –
each side is scaled by a factor of ½.
(More on this later.)
R
U
V
T
W
S



Guided Practice:
The midpoints of a triangle are X (–2, 5), Y
(3, 1), and Z (4, 8). Find the coordinates of
the vertices of the triangle.
1. Plot the midpoints on a coordinate
plane.
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

Guided Practice: continued
Connect the midpoints to form the
midsegments XY , YZ , and XZ .
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

Guided Practice: continued
Calculate the slope of each midsegment.

Calculate the slope of
m=
m=
y 2 - y1
x2 - x1
(1) - (5)
(3) - (-2)

4
5
The slope of
XY
.
Slope formula
Substitute (–2, 5) and (3, 1)
for (x1, y1) and (x2, y2).
Simplify.
XY
is
m=-
4
5
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
Guided Practice: continued

Calculate the slope of YZ .
m=
m=
y 2 - y1
Slope formula
x2 - x1
(8) - (1)
Substitute (3, 1) and (4, 8) for
(x1, y1) and (x2, y2).
(4) - (3)
7
m= =7
1

The slope of
Simplify.
YZ
is 7.
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
Guided Practice: continued

Calculate the slope of XZ .
m=
m=
y 2 - y1
Slope formula
x2 - x1
(8) - (5)
Substitute (–2, 5) and (4, 8)
for (x1, y1) and (x2, y2).
(4) - (-2)
3 1
m= =
6 2

The slope of
Simplify.
XZ
is
1
2
.
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

Guided Practice:
Draw the lines that contain the
midpoints.


The endpoints of each midsegment are
the midpoints of the larger triangle.
Each midsegment is also parallel to the
opposite side.
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
Guided Practice: continued

The slope of
is
XZ
1
2.
From point Y, draw
a line that has a
slope of 1 .

2
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
Guided Practice: continued

The slope of YZ is 7.
From point X, draw
a line that has a slope
of 7.

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
Guided Practice: continued
4
The slope of XY is - 5 .
 From point Z,
draw a line that 4
has a slope of - 5

The intersections of the
lines form the vertices
of the triangle.

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
Your turn
Determine the vertices of the triangle.
 The vertices of the triangle are (–3, –2), (9, 4),
and (–1, 12), as shown on the following slide.
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
This is all for today!!