Transcript Similar Polygons - William H. Peacock, LCDR USN, Ret

Similar Polygons
Section 7-2
Objective
• Identify similar polygons.
Key Vocabulary
• Similar polygons
• Similarity ratio
• Scale factor
Theorems
• 7.1 Perimeters of Similar Polygons
What is Similarity?
Similar Triangles
Not Similar
Similar
Similar
Not Similar
Similarity
Figures that have the
same shape but not
necessarily the same size
are similar figures. But
what does “same shape
mean”? Are the two
heads similar?
Similarity
Similar shapes can be thought of as
enlargements or reductions with no
irregular distortions.
– So two shapes are similar if one can be
enlarged or reduced so that it is congruent to
the original.
Similar Polygons
• When polygons have the same shape but
may be different in size, they are called
similar polygons.
• We express similarity using the symbol, ~.
(i.e. ΔABC ~ ΔPRS)
Example - Similar Polygons
Figures that are similar (~) have the same shape but not
necessarily the same size.
Similar Polygons
• The order of the vertices in a similarity
statement is very important. It identifies the
corresponding angles and sides of the polygons.
ΔABC ~ ΔPRS
A  P, B  R, C  S
AB = BC = CA
PR RS
SP
Similar Polygons
Two polygons are similar polygons iff the corresponding
angles are congruent and the corresponding sides are
proportional.
N
C
Similarity Statement:
M
CORN ~ MAIZ
Corresponding Angles:
C  CM
O  A
R  I
N  Z
O
R
N
Z
A
M
O Proportionality:
Statement of
R
CO  OR  RN  NC
MA AI IZ ZM
A
I
IMPORTANT
Writing a similarity statement is like writing a
congruence statement—be sure to list
corresponding vertices in the same order.
Similarity and Congruence
• If two polygons are congruent, they are
also similar.
• All of the corresponding angles are
congruent, and the lengths of the
corresponding sides have a ratio of 1:1.
Example 1
• If ΔABC ~ ΔRST, list all pairs of congruent angles and
write a proportion that relates the corresponding sides.
Example 1
• Use the similarity statement.
ΔABC ~ ΔRST
Answer:
Congruent Angles: A  R, B  S, C  T
Your Turn:
• If ΔGHK ~ ΔPQR, determine which of the following
similarity statements is not true.
A. HK ~ QR
B.
C. K ~ R
D. GHK ~ QPR
Example 2
PRQ ~ STU.
a. List all pairs of congruent angles.
b. Write the ratios of the corresponding
sides in a statement of proportionality.
c. Check that the ratios of
corresponding sides are equal.
SOLUTION
a. P  S, R  T, and Q  U.
b. ST = TU = US
PR
RQ
QP
c. ST = 8 = 4 , TU = 16 = 4 , and US = 12 = 4 .
PR
10
20
QP
15
5 RQ
5
5
4
The ratios of corresponding sides are all equal to .
5
Example 3
Determine whether the triangles are similar. If they are
similar, write a similarity statement and find the scale
factor of Figure B to Figure A.
SOLUTION
1. Check whether the corresponding angles are
congruent.
From the diagram, you can see that G  M,
H  K, and J  L. Therefore, the corresponding
angles are congruent.
Example 3
2. Check whether the corresponding side lengths are
proportional.
MK
=
GH
KL
=
HJ
12
12 ÷ 3
4
9 = 9÷3 = 3
16
16 ÷ 4
4
=
=
12 ÷ 4
3
12
All three ratios are equal,
so the corresponding side
lengths are proportional.
LM 20
20 ÷ 5
4
=
=
=
JG
15 ÷ 5
3
15
ANSWER
By definition, the triangles are similar.
GHJ ~ MKL.
4
The scale factor of Figure B to Figure A is .
3
Your Turn:
Determine whether the polygons are similar. If they are
similar, write a similarity statement and find the scale
factor of Figure B to Figure A.
1.
ANSWER
yes; XYZ ~ DEF;
ANSWER
no 9 ≠ 12
6 10
2.
3
2
Example 4a
• A. MENUS Tan is designing a new menu for the restaurant where he
works. Determine whether the size for the new menu is similar to
the original menu. If so, write the similarity statement and scale
factor. Explain your reasoning.
Original Menu:
New Menu:
Example 4a
Step 1
Compare corresponding angles.
Original
Since all angles of a rectangle
are right angles and right angles
are congruent, corresponding
angles are congruent.
Step 2
Compare corresponding sides.
New
Example 4a
Answer: Since corresponding sides are not
proportional, ABCD is not similar to
FGHK. So, the menus are not similar.
Example 4b
• B. MENUS Tan is designing a new menu for the restaurant where he
works. Determine whether the size for the new menu is similar to
the original menu. If so, write the similarity statement and scale
factor. Explain your reasoning.
Original Menu:
New Menu:
Example 4b
Step 1
Compare corresponding angles.
Original
Since all angles of a rectangle
are right angles and right angles are
congruent, corresponding angles are
congruent.
Step 2
Compare corresponding sides.
New
Example 4b
Answer: Since corresponding sides are
proportional, ABCD ~ RSTU. So the menus
4
__
are similar with a scale factor of 5 .
Your Turn:
A. Thalia is a wedding planner who Original:
is making invitations. Determine
whether the size for the new
invitations is similar to the original
invitations used. If so, choose the
correct similarity statement and
scale factor.
A.
BCDE ~ FGHI, scale factor =
B.
BCDE ~ FGHI, scale factor =
C.
BCDE ~ FGHI, scale factor =
D.
BCDE is not similar to FGHI.
1
2
4
5
3
8
New:
Your Turn:
B. Thalia is a wedding planner who is
making invitations. Determine
Original:
whether the size for the new
invitations is similar to the original
invitations used. If so, choose the
correct similarity statement and
scale factor.
1
1
__
A. BCDE ~ WXYZ, scale factor =
2
2
4
B. BCDE ~ WXYZ, scale factor = 4 __
5
5
3
C. BCDE ~ WXYZ, scale factor = 3 __
8
8
D. BCDE is not similar to WXYZ.
New:
Example 5a
A. The two polygons are similar. Find x.
Use the congruent angles
to write the corresponding
vertices in order.
polygon ABCDE ~ polygon RSTUV
Example 5a
Write proportions to find x.
Similarity proportion
Cross Products Property
Multiply.
Divide each side by 4.
Answer: x =
__
9
2
Example 5b
B. The two polygons are similar. Find y.
Use the congruent angles
to write the corresponding
vertices in order.
polygon ABCDE ~ polygon RSTUV
Example 5b
Similarity proportion
AB = 6, RS = 4, DE = 8, UV = y + 1
Cross Products Property
Multiply.
Subtract 6 from each side.
Divide each side by 6 and
simplify.
Answer: y =
__
13
3
Your Turn:
A. The two polygons are
similar. Solve for a.
A. a = 1.4
B. a = 3.75
C. a = 2.4
D. a = 2
Your Turn:
B. The two polygons are
similar. Solve for b.
A. 1.2
B. 2.1
C. 7.2
D. 9.3
Identifying Similar Triangles
• When only two congruent angles of a
triangle are given, remember that you
can use the Third Angles Theorem to
establish that the remaining
corresponding angles are also congruent.
• Example:
Example 6
Determine whether the pair of figures is similar.
Justify your answer.
T
Thus, all the corresponding angles are congruent.
Example 6
T
Now determine whether corresponding sides are
proportional.
The ratios of the measures of the corresponding sides
are equal.
Answer: The ratio of the measures of the corresponding
sides are equal and the corresponding angles
are congruent, so
Your Turn:
Determine whether the pair of figures is similar.
Justify your answer.
a.
Your Turn:
Answer: Both triangles are isosceles with base angles
measuring 76º and vertex angles measuring 28º.
The ratio of the measures of the corresponding
sides are equal and the corresponding angles
are congruent,
Your Turn:
Determine whether the pair of figures is similar.
Justify your answer.
b.
Answer: Only one pair of angles are congruent, so the
triangles are not similar.
Scale Factor
• In similar polygons, the
ratio of two corresponding
sides is called a scale factor.
• The scale factor depends on
the order of comparison.
• What is the scale factor of
the similar polygons shown?
CORN
N
8
C
4
5
O
6
R
Z
12
M
MAIZ
6
4 2
scale factor of CORN to MAIZ is 
6 3
6 3
A
scale factor of MAIZ to CORN is 
4 2
7.5
9
I
Scale Factor
• The scale factor between two similar
polygons is sometimes called the similarity
ratio.
• Scale factors are usually given for models
of real-life objects.
Example 7
An architect prepared a 12-inch model of a skyscraper
to look like a real 1100-foot building. What is the scale
factor of the model compared to the real building?
Before finding the scale factor you must make sure that
both measurements use the same unit of measure.
1100(12) = 13,200 inches
Scale factor
Example 7
Answer: The ratio comparing the two heights is
The scale factor is
which means that the model is
of the real skyscraper.
,
the height
Your Turn:
A space shuttle is about 122 feet in length. The Science
Club plans to make a model of the space shuttle with a
length of 24 inches. What is the scale factor of the
model compared to the real space shuttle?
Answer:
Example 8
The two polygons are similar. Find the scale factor of
polygon ABCDE to polygon RSTUV.
Example 8
The scale factor is the ratio of the lengths of any two
corresponding sides.
Answer:
Your Turn: Your Turn:
The two polygons are similar.
a. Write a similarity statement. Then find a, b, and ZO.
Answer:
;
b. Find the scale factor of polygon TRAP to polygon
Answer:
.
Example 9
Rectangle WXYZ is similar to rectangle PQRS
with a scale factor of 1.5. If the length and width
of rectangle PQRS are 10 meters and 4 meters,
respectively, what are the length and width of rectangle
WXYZ?
Write proportions for finding side measures. Let one long
side of each WXYZ and PQRS be
and one
short side of each WXYZ and PQRS be
Example 9
Answer:
Your Turn:
Quadrilateral GCDE is similar to quadrilateral JKLM with
a scale factor of
If two of the sides of GCDE
measure 7 inches and 14 inches, what are the lengths of
the corresponding sides of JKLM?
Answer: 5 in., 10 in.
Example 10
The scale on the map of a city is
inch equals 2 miles.
On the map, the width of the city at its widest point is
inches. The city hosts a bicycle race across town at its
widest point. Tashawna bikes at 10 miles per hour. How
long will it take her to complete the race?
Every
equals 2 miles. The
distance across the city at its widest point is
Example 10
Create a proportion relating the measurements to
the scale to find the distance in miles. Then use
the formula
to find the time.
Solve
Cross products
Divide each side by 0.25.
The distance across the city is 30 miles.
Example 10
Divide each side by 10.
It would take Tashawna 3 hours to bike
across town.
Answer: 3 hours
Your Turn:
An historic train ride is planned between two landmarks
on the Lewis and Clark Trail. The scale on a map that
includes the two landmarks is 3 centimeters = 125 miles.
The distance between the two landmarks on the map is
1.5 centimeters. If the train travels at an average rate of
50 miles per hour, how long will the trip between the
landmarks take?
Answer: 1.25 hours
Perimeters of Similar Polygons
• In similar polygons, the ratio of any two
corresponding lengths is proportional to
the scale factor between them.
• This leads to the following theorem about
the perimeters of two similar polygons.
Theorem 7.1 - Perimeters of Similar
Polygons
If two polygons are similar, then their
perimeters are proportional to the scale
factor between them.
Example 11
If ABCDE ~ RSTUV, find the scale factor of ABCDE to RSTUV
and the perimeter of each polygon.
Example 11
The scale factor ABCDE to RSTUV
AE
is
VU
___
4
or .
7
Write a proportion to find the length of DC.
Write a proportion.
4(10.5)= 7 ● DC
6 = DC
Cross Products Property
Divide each side by 7.
Since DC  AB and AE  DE, the perimeter of ABCDE is 6 + 6 +
6 + 4 + 4 or 26.
Example 11
Use the perimeter of ABCDE and scale
factor to write a proportion. Let x
represent the perimeter of RSTUV.
Theorem 7.1
Substitution
4x = (26)(7)
x = 45.5
Cross Products Property
Solve.
Example 11
Answer: The perimeter of ABCDE is 26 and the perimeter of
RSTUV is 45.5.
Your Turn:
If LMNOP ~ VWXYZ, find the perimeter
of each polygon.
A. LMNOP = 40, VWXYZ = 30
B. LMNOP = 32, VWXYZ = 24
C. LMNOP = 45, VWXYZ = 40
D. LMNOP = 60, VWXYZ = 45
Example 12
RST ~ GHJ.
Find the value of x.
SOLUTION
Because the triangles are similar, the corresponding
side lengths are proportional. To find the value of x,
you can use the following proportion.
GH
=
RS
15
=
10
JG
TR
9
x
15 · x = 10 · 9
Write proportion.
Substitute 15 for GH, 10 for RS,
9 for JG, and x for TR.
Cross product property
Example 12
15x = 90
15x 90
=
15
15
x=6
Multiply.
Divide each side by 15.
Simplify.
Example 13
The outlines of a pool and the patio around the pool
are similar rectangles.
a. Find the ratio of the length
of the patio to the length of
the pool.
b. Find the ratio of the
perimeter of the patio to
the perimeter of the pool.
SOLUTION
a. The ratio of the length of the patio to the length of
the pool is
length of the patio
48 feet
48 ÷ 16
3
=
=
= .
32 ÷ 16
2
length of pool
32 feet
Example 13
b. The perimeter of the patio is 2(24) + 2(48) = 144 feet.
The perimeter of the pool is 2(16) + 2(32) = 96 feet.
The ratio of the perimeter of the patio to the
perimeter of the pool is
perimeter of patio
perimeter of pool
=
144 feet
144 ÷ 48 3
=
= .
96 ÷ 48 2
96 feet
Your Turn:
In the diagram, PQR ~ STU.
1. Find the value of x.
ANSWER
18
2. Find the ratio of the perimeter of
STU to the perimeter of PQR.
ANSWER
1
2
Assignment
• Pg. 368 – 371: #1 – 37 odd