17-3: Properties of Similar Figures

17-3: Properties of Similar Figures
Objectives:
1. To define and use
similar polygons
Assignment:
• P. 254: 1-11
• P. 255-265: 7-11
• Challenge Problems
Objective 1
You will be able to
define and use similar
polygons
Guggenheim Museum
Building big stuff
can be
expensive. So to
work out details,
artists and
architects usually
build scale
models.
(Frank Lloyd Wright is a giant)
Guggenheim Museum
Building big stuff
can be
expensive. So to
work out details,
artists and
architects usually
build scale
models.
Guggenheim Museum
A scale model is
similar to the
actually object
that is to be built.
And that does not
mean that they
are kind of alike.
Guggenheim Museum
A scale model is
similar to the
actually object
that is to be built.
And that does not
mean that they
are kind of alike.
Similarity
Figures that have the
same shape but not
necessarily the same
size are similar figures.
But what does “same
shape mean”? Are Mr.
Noel’s heads similar?
Similarity
Similar shapes can be thought
of as enlargements or
reductions with no irregular
distortions.
So two shapes are similar
if one can be enlarged or
reduced so that it is
congruent to the original.
Investigation 1
Use the “What are
Similar Polygons”
handout to discover a
mathematical
definition of similar
polygons. The two
illustrations represent
the enlargement and
reduction of an
“impossible” 3-D solid.
Similar Polygons
Two polygons are similar polygons iff the corresponding
angles are congruent and the corresponding sides are
proportional.
Similarity Statement:
N
C
𝐶𝑂𝑅𝑁~𝑀𝐴𝐼𝑍
Corresponding Angles:
N
C
∠𝐶 ≅ ∠𝑀 ∠𝑂 ≅ ∠𝐴
∠𝑅 ≅ ∠𝐼 ∠𝑁 ≅ ∠𝑍
O
M
Z
R
A
O
Statement
of Proportionality:
R
𝐶𝑂 𝑂𝑅 𝑅𝑁 𝑁𝐶
=
=
=
𝑀𝐴 𝐴𝐼
𝐼𝑍
𝑍𝑀
A
I
Example 1
Use the definition of similar polygons to find
the measure of x and y, assuming SMAL ~
BIGE.
Example 2
When asked to find
the length of 𝐷𝐸
given that the
triangles are
similar, Kenny
says 10. Explain
what is wrong
with Kenny’s
reasoning.
D
A
6
5
F
C
3
8
B
E
10
Scale Factor
C
In similar polygons,
the ratio of two
corresponding sides
is called a scale
factor.
N
8
4
5
O
6
R
Z
12
M
6
4
6
Scale factor = =
2
3
7.5
A
9
I
Corresponding Lengths
Corresponding Lengths in Similar
Polygons
If two polygons are similar, then the ratio of
any two corresponding lengths in the
polygons is equal to the scale factor of the
similar polygons.
Sides
Altitudes
Medians
Midsegments
Example 3
In the diagram
Δ𝑇𝑃𝑅~Δ𝑋𝑃𝑍. Find
the length of the
altitude 𝑃𝑆.
Similarity Transformations
Transformations in which the pre-image and
image are similar are called
similarity transformations.
Translation
Reflection
Rotation
Dilation
Investigation 2
Similarity
transformations
preserve angle
measures and
create proportional
lengths, but what do
they do to the
perimeter and area
of a polygon?
Similarity Relationships
For two shapes with a scale factor of 𝑎: 𝑏,
each of the following relationships will be
true:
Perimeter
Linear
Units
𝑎: 𝑏
Area
Square
Units
𝑎2 : 𝑏2
Volume
Cubic
Units
𝑎3 : 𝑏3
Example 4
In the diagram, ABCDE ~ FGHJK. Find the
perimeter of ABCDE.
F
15
G
9
A
10
H
B
C
18
12
E
D
K
15
J
Example 5
In the diagram, ABCDE ~ FGHJK. Find the
area of ABCDE.
17-3: Properties of Similar Figures
Objectives:
1. To define and use
similar polygons
Assignment:
• P. 254: 1-11
• P. 255-265: 7-11
• Challenge Problems