Similarity
*Same shape but not necessarily the same size
Similar Triangles
If two shapes are similar, one is an enlargement of the other. This means that the two
shapes will have the same angles and their sides will be in the same proportion (e.g. the
sides of one triangle will all be 3 times the sides of the other etc.).
angle A = angle D
angle B = angle E
angle C = angle F
AB/DE = BC/EF = AC/DF = perimeter of ABC/ perimeter of DEF
Two triangles are similar if:
1) 3 angles of 1 triangle are the same as 3 angles of the other
or 2) 3 pairs of corresponding sides are in the same ratio
or 3) An angle of 1 triangle is the same as the angle of the other triangle and the sides
containing these angles are in the same ratio.
Example:
In the above diagram, the triangles are similar. EF = 6cm and BC = 2cm . What is the
length of DE if AB is 3cm?
EF = 3BC, so DE = 3AB = 9cm.
Triangles are similar if their corresponding (matching) angles are equal and the
ratio of their corresponding sides are in proportion.
Create a proportion
matching the
corresponding sides.
Find x:
Two possible answers:
Small
triangle on
top:
Large triangle
on top:
x = 20
x = 20
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.
Strategies for
Math A
Dealing with Similar
Triangles
Triangles are similar if their corresponding (matching) angles are equal and the
ratio of their corresponding sides are in proportion.
There are many different types of
problems that involve similar
triangles. And, fortunately, there are
many different ways to arrive at an
answer.
Keep an open mind!
Remember that there is
more than one way to
arrive at an answer!
Let's look at some strategies for arriving at answers!
The easiest problems dealing with similar triangles are those that involve two
separate triangles.
Find x:
Create a proportion
matching the
corresponding sides.
Two possible answers:
Small
triangle on
top:
Large triangle
on top:
x = 20
x = 20
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.
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.
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
Find BE:
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!!
Read carefully to see WHAT you are
supposed to find. This problem asks you
to find BE.
Here are two solutions.
Use FULL sides of the Use the rule related to
triangles, cross
parallel lines, cross
multiply and
multiply and solve.
solve.
4x + 36 = 12x
36 = 8x
4.5 = x
Find EC:
36 = 8x
4.5 = x
This problem asks you to find EC.
Here are two solutions:
Use FULL sides of the Use the rule related to
triangles, cross multiply parallel lines, cross
and
multiply and solve.
solve.
32 + 4x = 80
4x = 48
x = 12
Find x:
4x = 48
x = 12
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:
x=5
HINT:
If the triangles which are on top of one another
are causing you problems,
redraw the triangles separating them into two
separate figures.
http://www.regentsprep.org/Regents/Math/similar/PracSim.htm