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Triangles are similar
if their corresponding (matching) angles are equal and the ratio
of their corresponding sides are in proportion. |
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There
are many different types of problems that
involve similar triangles. And,
fortunately, there are many different ways to
arrive at an answer. |
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Keep
an open mind! Remember that there is
more than one way to arrive at an answer! |
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Let's
look at some strategies for arriving at answers!
The easiest
problems dealing with similar triangles are those that involve two
separate triangles.
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Find
x:
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Create
a proportion matching the corresponding sides.
Two
possible answers:
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Small triangle on
top:
x = 20
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Large triangle on
top:
x = 20
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HINT:
These two triangles are sitting such that their
corresponding parts are in the same position in each
triangle. If
the triangles are not sitting in this manner, you can match the corresponding sides by looking across from
the angles which are equal in each triangle. |
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Many
problems involving similar triangles have one triangle
ON
TOP OF
another triangle. Since DE is marked to be parallel
to AC, we know that we have angle BDE equal to angle DAC
(corresponding angles). Angle B is shared by both
triangles, so the triangles are similar by AA. |
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There
are two ways to attack this problem. |
Use
FULL sides of the two triangles when dealing with
the problem. Do not use DA or EC since they are not
sides of triangles.
EASIEST
METHOD TO USE |
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Use
a rule relating to parallel lines, which says that if a
line is parallel to one side of a triangle, it divides the
other sides proportionately.
EASY TO FORGET!! |
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Find
BE:
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Read
carefully to see WHAT you are supposed to find.
This problem asks you to
find BE.
Here
are two solutions. |
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Use FULL sides of the triangles, cross
multiply and solve.
4x + 36 = 12x
36 = 8x
4.5 = x |
Use the rule related to parallel
lines, cross multiply and solve.
36 = 8x
4.5 = x |
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Find
EC:
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This
problem asks you to
find
EC.
Here
are two solutions: |
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Use FULL sides of the triangles, cross
multiply and solve.
32 + 4x = 80
4x = 48
x = 12 |
Use the rule related to parallel
lines, cross multiply and solve.
4x = 48
x = 12 |
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Find
x:
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CAREFUL!!!
This problem MUST use the full sides of triangles as a
solution. The parallel rule does not work
here. The problem asks you to
find
x.
Here
is the solution: |
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x = 5 |
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HINT: |
If
the triangles which are
on
top of
one another are causing you problems,
redraw the triangles separating them into two separate
figures. |
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