Download Chapter 7 Similarity

Chapter 7 Similarity
7.1 Ratios and Proportions

Ratio



A comparison of two quantities
A
A to B, A:B, or
Proportion


B
A Statement that two ratios are equal
Ex. A C

B D
Properties of Proportions

If
a c

b d
then
1
ad  bc
2
b
d

a
c

3
a
b

c
d

4 ab


b
cd

d
Cross-Product Property



The product of the
extremes is equal to
the product of the
means
means
a:b  c: d
extremes
Examples

7.1 Examples
7-2 Similar Polygons


chapter 7.2 similarity.gsp
Two polygons are similar if
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

Corresponding angles are congruent
Corresponding sides are proportional
Similarity ratio


Ratio of the lengths of the corresponding
sides
Practice with clickers
Golden Ratio


http://www.youtube.com/watch?feature
=player_detailpage&v=ReJOK8RMzPE
http://www.youtube.com/watch?list=PL
5E4F2F128AFE5A3D&feature=player_de
tailpage&v=085KSyQVb-U#t=4s
Golden Ratio


1:1.618
Also known as the golden rectangle
7.2 Examples

Examples
7.3 Proving Triangles Similar

AA Similarity Postulate

If two angles of one triangle are congruent
to two angles of another triangle, then the
triangles are similar.
Similarity Theorems

SAS Similarity


If an angle of one triangle is congruent to
an angle of a second triangle, and the
sides including the two angles are
proportional, then the triangles are similar.
SSS Similarity

If the corresponding sides of two triangles
are proportional, then the triangles are
similar.
Examples

Examples
7.4 Similarity in Right Triangles

Theorem 7-3

The altitude to the hypotenuse of a right
triangle divides the triangle into two
triangles that are similar to the original
triangle and to each other.
Geometric Mean
The geometric mean of a and b is
a x

x b
Example

Find the geometric mean of 4 and 18
Corollaries to Theorem 7-3

Corollary 1


Piece of hypotenuse = Altitude
Altitude
Other piece of hypotenuse
Corollary 2

Piece of hypotenuse = Leg
Leg
Whole Hypotenuse
7.5 Proportions in Triangles

Theorem 7-4 Side Splitter Theorem

If a line is parallel to one side of a triangle and
intersects the other two sides, then it divides
those sides proportionally.
B

BD
AD
=
BE
E
CE
D
C
A
Corollary to Theorem 7-4

If three parallel lines
intersect two
transversals, then the
segments intercepted
on the transversals are
proportional.
a c

b d
c
a
b
dd
Theorem 7-5 example
Triangle-Angle-Bisector Theorem

If a ray bisects an angle of a triangle, then it divides
the opposite side into two segments that are
proportional to the other two sides of the triangle.
B
m
n

p
o
m
p

n
o
p
m
C
D
A
n
o