Transcript Similar Triangles - C on T ech Math : : An application

Similar Triangles
Tutorial 12g
Similarity
When people say that two things are similar, usually they
mean that the things are alike. In geometry, similar figures
are alike in a very specific way.
Figures that are similar have the same shape but may or may
not have the same size.
Here are three photos: the original photo, the enlargement of
the original, and a reduction of the original photo. Because all
three photos show the same shape, the figures are similar.
Similar Polygons
Two polygons are in correspondence when consecutive sides
and vertices of one are matched with consecutive sides and
vertices of the other.
Similar Polygons
For two polygons to be similar they must have:
 1. Corresponding (or matching) angles that have the same measure.
In the two rectangles below, all four angles in each are right angles, therefore
they meet the first criteria of corresponding angles having the same measure.
 2. Corresponding (or matching) sides that are proportional.
Are all corresponding sides proportional?
8 12
Does 10  15 ? Yes, these ratios are equal.
Therefore these two rectangles also meet the second
criteria of corresponding sides being proportional.
10 in.
8 in.
15
15 in.
12 in.
The ratios of the lengths of
corresponding sides are equal.
Similar Triangles
Triangles have some special relationships between sides
and angles when they are similar. It can be very helpful in
mathematics to determine when two triangles are similar.
If two triangles are similar, it becomes easier to learn more
characteristics of the two triangles.
It is not always necessary to prove both criteria, that all 3
corresponding angles have equal measures and all
corresponding sides are proportional to determine whether
or not two triangles are similar.
The following mathematical rules give us two short cuts:
Similar Triangles
cont. . .
Angle Angle (AA) Similarity Postulate
If two angles of one triangle are congruent to
two angles of another triangle, then the two
triangles are similar.
D
A
Since A  D & B  E,
then ABC is similar to DEF
43
43
B
C
E
F
“” means “is similar to”.
So we can write the similarity using symbols: ABC  DEF.
Similar Triangles
cont. . .
Angle Angle (AA) Similarity Postulate
If two angles of one triangle are congruent to
two angles of another triangle, then the two
triangles are similar.
D
A
Since A  D & B  E,
then ABC is similar to DEF
43
43
B
C
E
F
“” means “is similar to”.
So we can write the similarity using symbols: ABC  DEF.
Similar Triangles
cont. . .
Since these two triangles are similar, we can use
proportions to find the length of the missing side.
D
Since the two triangles are similar,
the ratios of the corresponding sides are
proportional.
Therefore:
6 8

x 12
So, 8x = 6•12
8x = 72
x=9
Lets check our answer:
6 8
Does  ? YES!
9 12
43
A
43
12 in.
8 in.
B
6 in.
C
E
x
F
Similar Triangles
cont. . .
SSS Similarity Theorem
If the sides of two triangles are in
proportion, then the triangles are similar.
Example:  POM   LMN, what are the lengths of the missing sides?
L
7
Solution:
?
10.5
?
O
7
PM = 7
PO = 6
LN = 10.5
MN = 12
LM = ?
MO = ?
7
12
M
6
P
10 .5
N
10 .5


6
LM

MO
12
6
7
LM
10 .5
7(LM) = 10.5  6
7(LM) = 63
7( LM ) 63

7
7
LM = 9

MO
12
10.5(MO) = 12  7
10.5(MO) = 84
10 .5( MO )
84

10 .5
10 .5
MO = 8