Transcript Chapter 10

Chapter 10:
Similarity
BY: JUSTIN KIM
&
KEVIN PRAETORIUS
Lesson 1:
Ratio and Proportion
 The ratio of the number a to the number b is the
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number .
A proportion is an equality between ratios
A proportion can be represented symbolically as
=
a is the first term of a proportion
b is the second
c is the third
d is the fourth
Lesson 1 Continued
 The 2nd and 3rd terms are the means
 1st and 4th terms are the extremes
 The product of the means equals the product of the
extremes
 If you take = , you can cross multiply to get
ad=cb
 If the means are equal, they are a geometric mean
 The number b is the geometric mean between the
numbers a and c if =
Lesson 2: Similar Figures
 Two triangles are similar iff there is a
correspondence between their vertices such that
their corresponding sides are proportional and
their corresponding angles are equal.
 The center of a dilation is the point at which a
shape is dilated
 The magnitude of a dilation is the relative size of
an image compared with the original
Lesson 3: The Side Splitter Theorem
 The Side Splitter Theorem- If a line parallel to one
side of a triangle intersects the other two sides in
different points, it divides the sides in the same
ratio.
 Corollary- If a line parallel to one side of a triangle
intersects the other two sides in different points, it
cuts off segments proportional to the sides.
The Side Splitter Theorem
A
XZ is || to BC therefore it
is a side splitter to
ABC…
X
Therefore
Z
Or…
=
=
Which is the Corollary
B
C
Lesson 4: The AA similarity Theorem
 The Angle Angle (AA) Theorem- If two angles of
one triangle are equal to two angles of another
triangle, the triangles are similar.
 Corollary- Two triangles similar to a third triangle
are similar to each other
AA similarity
D
A
ABC ~ DEF
A
If
B
C
B
E
F
C
A ~ B and B ~ C, then A ~ C
Lesson 5: Proportions
and Dilations
 Corresponding altitudes of triangles are altitudes
that are drawn from corresponding vertices.
 Corresponding altitudes of similar triangles have the
same ratio as that of the
E
corresponding sides.
B
A
C
D
F
Lesson 6: Perimeters and Areas
of Similar Figures
 The ratio of the perimeters of two similar polygons is
equal to the ratio of their corresponding sides.
 The ratio of the areas of two similar polygons is equal
to the square of the ratio of their corresponding
sides.
 So, if the ratio of the sides of two similar triangles is
, the ratio of their perimeters is and the ratios of
their areas is
Additional Lesson: The Angle Bisector Theorem
 The angle bisector theorem states that an angle
bisector in a triangle divides the opposite side into
segments that have the same ratio as the other two
sides.
As a
proportion,
B
X
=
A
C
Extra Homework Problems
 In the extra homework problems we had to use the
Side Splitter Theorem, AA Similarity Theorem, and
Angle Bisector Theorem to find the areas and
perimeters of triangles.
 One of the problems asks you to
find length x given that the line
drawn is an angle bisector of
the triangle.
Extra Homework Problems
First, set up a proportion.
=
Next, cross multiple the
proportion.
36 = 10x - 30
Now simplify and
solve for x.
66 = 10x
6.6 = x