The use of similar triangles appears in the solutions to many different types of mathematical problems. The good news is that there are often several different ways to arrive at answers to these problems. Keep an open mind, and be on the look out for the possibility of several different solution methods.
openmind

Consider the following strategies using similar triangles:

Situation 1: Two separate, marked triangles
Find x.

strat1
The marked diagrams tell you the triangles are similar (AA).

Create a proportion of the corresponding sides to solve for x.

Consider these possible solution methods:
Large triangle
in numerators:

strat1T
x = 14
Small triangle
in numerators
strat1T2
x = 14
Group triangles
in ratios
strat1T3
x = 14
 beware The triangles seen in this problem are positioned such that their corresponding parts are in the same positions in each triangle. If the triangles are not positioned in this manner, you can match the corresponding sides by looking across from the angles which are marked to be congruent (or known to be congruent) in each triangle.

 

 

Situation 2: Overlapping triangles
Is ΔABC similar to ΔDBE?
stat2

Many problems involving similar triangle have one triangle on top of (overlapping) the other.
In this diagram, it is indicated that stat2T1
.

Since ∠A is congruent to ∠BDE (corresponding ∠s from the || lines), and ∠B is shared by both triangles, we have similarity of the triangles by AA.

So,
YES, the triangles are similar.

Now, if you are asked to find a missing side (or segment) in this overlapping situation,
you have two solution methods to consider:
Use FULL sides from the two triangles.
Do not use DA or EC in your proportion since they are not the lengths of SIDES of the triangles.

When in doubt, use this method!

Use Side Splitter Theorem:
"If a line is parallel to one side of a triangle, and intersects the other two sides, the line divides these two sides proportionally."
This method may not apply in all situations.
The choice of which method to use will depend upon the given information regarding the sides, what you are asked to find, and which method you feel most secure in using.

 


Situation 3: Find side of triangle given lengths that are not sides of triangles
Find BE.

stat3
Always read carefully to see WHAT you are supposed to find. This question asks you to find BE, which is a length of a side of a triangle.

From Situation 2, we know these triangles are similar.

Consider these two possible solution methods,
letting BE = x:
Use FULL sides
of the triangles:

(do not use 12 or 15
as they are not sides of a triangle)

stat3T1
6x + 90 = 18x
90 = 12x
7.5 = x
Use Side Splitter Theorem:
(since we have "non-triangle sides" segments)
stat3T2
90 = 12x
7.5 = x

 

 

Situation 4: Find segment that is not the side of a triangle
Find EC.

stat4
This question asks you to find EC, which is not a length of a side of a triangle.

From Situation 2, we know these triangles are similar.

Consider these two possible solution methods,
letting EC = x:
Use FULL sides
of the triangles:

(do not use 3 or x by itself
as they are not sides of a triangle)

stat4T2
135 + 9x = 180
9x = 45
x =
5
Use Side Splitter Theorem:
(since we have "non-triangle sides" segments)
stat4T#
9x = 45
x =
5

 

beware
Just when the "Side Splitter Theorem" method was looking like the easier method ...

Situation 5: BE CAREFUL here!!
Find DE.

stat5
This question MUST use the FULL sides method of solution. The Side Splitter Theorem will not work here!
While stat2T1, we are not given any information about the lengths of the segments formed by de3 on side bc.

From Situation 2, we know these triangles are similar.

The only solution method is to use FULL sides:
stat5T1
144 = 24x
6 = x



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