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Triangles are similar
if their corresponding (matching) angles are congruent (equal in
measure) 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 several different ways to
arrive at an answer. |
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Keep
an open mind! Remember that there
may be
more than one way to arrive at an answer! |
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Let's
look at some strategies for arriving at answers!
Style 1:
The similar triangles are 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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Style 2:
The similar triangles
overlap:
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Many
problems involving similar triangles have one triangle
ON
TOP OF
(overlapping)
another triangle.
Since
 is marked to be parallel
to  , we know that we have
<BDE congruent to <DAC
(by corresponding angles). <B is shared by both
triangles, so the two triangles are similar by AA. |
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There
are two ways to attack this type of 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 theorem relating to parallel lines, which says that If a
line is parallel to one side of a triangle, and intersects
the other two sides, the line divides these two sides proportionally.
EASY TO FORGET!! |
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Let's try some problems with this type of
diagram:
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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 letting BE = x. |
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Use FULL sides of the triangles, cross
multiply and solve.
4x + 36 = 12x
36 = 8x
4.5 = x |
Use the
theorem 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 letting EC = x: |
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Use FULL sides of the triangles, cross
multiply and solve.
32 + 4x = 80
4x = 48
x = 12 |
Use the
theorem 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 theorem does not work
here. The problem asks you to
find
x
where x is a FULL side.
Here
is the solution: |
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x = 5 |
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HINT: |
If
you have a hard time "seeing" what is happening in
overlapping triangles,
redraw the triangles as two separate figures. |
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