7.2 similar polygons - montgomery county · pdf fileidentify similar polygons in real-life ......
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7.2 Similar Polygons
Geometry
Mr. Peebles
Spring 2013
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Daily Learning Target (DLT)
Monday February 25, 2013 • “I can understand, apply, and remember to
identify similar polygons in real-life
problems.”
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Geometry Bell Ringer
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Geometry Bell Ringer
Answer: C
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Identifying similar polygons • When there is a correspondence between
two polygons such that their corresponding angles are congruent and the lengths of corresponding sides are proportional the two polygons are called similar polygons.
• Remember that geometric figures that are similar have the same shape and angles but have different sizes.
• In the next slide, ABCD is similar to EFGH. The symbol ~ is used to indicate similarity. So, ABCD ~ EFGH.
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Similar polygons
B
C
AD
F
G
E H
AB = =
EF
BC
FG =
CD
GH
DA
HE
AB =
EF
Polygon AB
Polygon EF
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Ex. 1: Writing Similarity Statements
• Pentagons JKLMN and STUVW are
similar. List all the pairs of congruent
angles. Write the ratios of the
corresponding sides in a statement of
proportionality. J K
L
M
N
S T
U
V
W
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Ex. 1: Writing Similarity Statements
J K
L
M
N
S T
U
V
W
Because JKLMN ~ STUVW, you can
write J S, K T, L U, M
V AND N W.
You can write the
proportionality statement
as follows:
KL
TU =
JK =
ST
MN
VW =
LM =
UV
NJ
WS
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Ex. 2: Comparing Similar Polygons
• Decide whether the figures are similar. If
they are similar, write a similarity
statement.
15
12
9
6X
W
Z
Y
10
8
6
4Q
P
S
R
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15
12
9
6X
W
Z
Y
10
8
6
4Q
P
S
RSOLUTION:
As shown, the corresponding
angles of WXYZ and PQRS
are congruent. Also, the
corresponding side lengths
are proportional.
WX
PQ =
15
10 =
3
2
XY
QR =
6
4 =
3
2
YZ
RS =
9
6 =
3
2
WX
PQ =
15
10 =
3
2 So, the two figures are
similar and you can write
WXYZ ~ PQRS.
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Ex. 3: Comparing Photographic
Enlargements
• POSTER DESIGN. You have been
asked to create a poster to advertise a
field trip to see the Liberty Bell. You
have a 3.5 inch by 5 inch photo that you
want to enlarge. You want the
enlargement to be 16 inches wide. How
long will it be? Hint… you will need to
set up a proportion comparing the length
and width of the picture and frame.
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Solution:
• To find the length of the enlargement,
you can compare the enlargement to the
original measurements of the photo.
16 in.
3.5 in. =
x in.
5 in.
x = 16
3.5 x∙ 5
x ≈ 22.9 inches
The length of the
enlargement will be about 23
inches.
5
3.5
Trip to Liberty Bell
March 24th,
Sign up
today!
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Theorem 8.1
• Theorem 8.1: If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding parts.
• If KLMN ~ PQRS, then
P
S
Q
RK
N
L
M
KL + LM + MN + NK
PQ + QR + RS + SP =
KL
PQ
LM
QR
MN
RS
NK
SP = = =
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Using similar polygons in real life
• If two polygons are similar, then the ratio
of lengths of two corresponding sides is
called the scale factor. In Example 2 on
the previous page, the common ratio of
is the scale factor of WXYZ to PQRS.
3
2
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Ex. 4: Using similar polygons
• The rectangular patio
around a pool is
similar to the pool as
shown. Calculate the
scale factor of the
patio to the pool, and
find the ratio of their
perimeters.
• The ratio is the
• Patio/Pool
16 ft 24 ft
32 ft
48 ft
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• Because the rectangles are
similar, the scale factor of
the patio to the pool is 48
ft: 32 ft. , which is 3:2 in
simplified form.
• The perimeter of the patio
is 2(24) + 2(48) = 144 feet
and the perimeter of the
pool is 2(16) + 2(32) = 96
feet The ratio of the
perimeters is
16 ft 24 ft
32 ft
48 ft 144
96
3
2 , or
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NOTE:
• Notice in Example 4 that the ratio of
perimeters is the same as the scale factor
of the rectangles. This observation is
generalized in the following theorem.
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Ex. 5: Using Similar Polygons
• Quadrilateral JKLM is
similar to PQRS. Find the
value of z.
Set up a proportion that contains PQ
15
10J
M L
K
Z
6
S
PQ
R
KL
QR
JK
PQ =
Write the proportion.
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Ex. 5: Using Similar Polygons
• Quadrilateral JKLM is
similar to PQRS. Find the
value of z.
Set up a proportion that contains PQ
15
10J
M L
K
Z
6
S
PQ
R
KL
QR
JK
PQ =
15
6
10
Z =
Z = 4
Write the proportion.
Substitute
Cross multiply and divide by 15.
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Assignment
Pages 376 (1-16)
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Closure
On the whiteboards, write down what it
means for polygons to be similar.
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Closure
On the whiteboards, write down what it
means for polygons to be similar.
Answer:
Remember that geometric figures that are
similar have the same shape and angles
but have different sizes.