Chapter 7 Similarity
Download
Report
Transcript 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 ab
b
cd
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
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