Geometry Bell Ringer
• 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
18 ft 27 ft
28 ft
42 ft
Geometry Bell Ringer
• 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
• Answer: 3/2 or 1.5
is the Scale Factor.
18 ft 27 ft
28 ft
42 ft
Daily Learning Target (DLT)
Tuesday February 26, 2013 • “I can understand, apply, and remember
how to identify similar polygons notably
in real-life problems.”
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.
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
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
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
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.
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.
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!
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 = = =
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
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
• 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
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.
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.
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.
Assignment
Pages 376 (1-16)
1. Angle JHY 10. ABCD ~ FGHE
2. Angle R 11. They are NOT Similar
3. Angle JXY 12. ∆ABC ≅ ∆FED
4. Side HY 13. x = 4, y = 3
5. Side JT 14. x = 20, y = 17.5, z = 7.5
6. Side HY 15. x = 16, y = 4.5, z = 7.5
7. They are NOT Similar 16. x = 6
8. QRST ~ XWZY y = 8
9. KLMJ ~ PQNO z = 10
Geometry Exit Quiz – 5 Points
• 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.
27 ft 36 ft
39 ft
52 ft