Similar Triangles I

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Similarity of Triangles
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In geometry, two polygons are similar when one is
a replica (scale model) of the other.
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Consider Dr. Evil and Mini Me from Mike Meyers
hit movie Austin Powers. Mini Me is supposed to
be an exact replica of Dr. Evil.
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The following are similar figures.
I
II
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The following are non-similar figures.
I
II
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Feefee the mother cat, lost her daughters, would
you please help her to find her daughters. Her
daughters have the similar footprint with their
mother.
Feefees footprint
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1.
Which of the following is similar to the above
triangle?
B
A
C
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Note One triangle is a scale model of the other
triangle.
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How do we know if two triangles are similar or
proportional?
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Triangles are similar () if corresponding angles
are equal and the ratios of the lengths of
corresponding sides are equal.
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Interior Angles of Triangles
The sum of the measure of the angles of a
triangle is 1800. Ð A Ð B ÐC 1800
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Example 6-1b
Determine whether the pair of triangles is
similar. Justify your answer.
Answer Since the corresponding angles have
equal measures, the triangles are similar.
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If the product of the extremes equals the product
of the means then a proportion exists.
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This tells us that ? ABC and ? XYZ are similar
and proportional.
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Q Can these triangles be similar?
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AnswerYes, right triangles can also be similar
but use the criteria.
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Do we have equality?
This tells us our triangles are not similar. You
cant have two different scaling factors!
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If we are given that two triangles are similar or
proportional what can we determine about the
triangles?
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The two triangles below are known to be similar,
determine the missing value X.
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In the figure, the two triangles are similar.
What are c and d ?
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In the figure, the two triangles are similar.
What are c and d ?
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MODELING A REAL-LIFE PROBLEM
GEOMETRY CONNECTION Two students are visiting
the mysterious statues on Easter Island in the
South Pacific. To find the heights of two statues
that are too tall to measure, they tried a
technique involving proportions. They measured
the shadow lengths of the statues at 200 P.M.
and again at 300 P.M.
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SOLUTION
They let a and b represent the heights of the two
statues. Because the ratios of corresponding
sides of similar triangles are equal, the
students wrote the following two equations.
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Draw Similar Rectangles ABCD and EFGH whose
lengths and widths are 16 and 12 and 12 and 9
respectively.
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12
16
9
12
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Two triangles are called similar if their
corresponding angles have the same measure.
?
?
?
?
?
?
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Two triangles are called similar if their
corresponding angles have the same measure.
?
Ratios of corresponding sides are equal.
C
A
?
a
c
?
?
?
?
b
B
a
A
b
B
c
C


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Mary is 5 ft 6 inches tall. She casts a 2 foot
shadow. The tree casts a 7 foot shadow. How tall
is the tree?
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Mary is 5 ft 6 inches tall. She casts a 2 foot
shadow. The tree casts a 7 foot shadow. How tall
is the tree?
Marys height
Trees height
Marys shadow
Trees shadow

x
5.5
2
7
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Mary is 5 ft 6 inches tall. She casts a 2 foot
shadow. The tree casts a 7 foot shadow. How tall
is the tree?
5.5
x
2
7

Marys height
Trees height
Marys shadow
Trees shadow

x
5.5
2
7
35
5.5
x
2
7

7 ( 5.5 ) 2 x 38.5 2 x
x 19.25
The height of the tree is 19.25 feet
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Example 6-2b
Find the missing measures if the pair of
triangles is similar.
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Example 6-2b
Answer The missing measure is 7.5.
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Example 6-2c
Answer The missing measures are 18 and 42.
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Example 6-2c
Answer The missing measure is 5.25.
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Example 6-3a
Shadows Richard is standing next to the General
Sherman Giant Sequoia three in Sequoia National
Park. The shadow of the tree is 22.5 meters, and
Richards shadow is 53.6 centimeters. If
Richards height is 2 meters, how tall is the
tree?
Since the length of the shadow of the tree and
Richards height are given in meters, convert the
length of Richards shadow to meters.
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Example 6-3a
Answer The tree is about 84 meters tall.
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Example 6-3b
Tourism Trudie is standing next to the Eiffel
Tower in France. The height of the Eiffel Tower
is 317 meters and casts a shadow of 155 meters.
If Trudies height is 2 meters, how long is her
shadow?
Answer The length of Trudies shadow is about
0.98 meter.
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Similarity of Triangles
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