Warm Up
Convert each measurement.
1. 6 ft 3 in. to inches
75 in.
2. 5 m 38 cm to centimeters
538 cm
Find the perimeter and area of each
polygon.
3. square with side length 13 cm
P = 52 cm, A =169 cm2
4. rectangle with length 5.8 m and width
2.5 m
P =16.6 m, A = 14.5 m2
A scale drawing represents an object as smaller than or larger than its actual
size. The drawing’s scale is the ratio of any length in the drawing
to the corresponding actual length. For example, on a map with a scale of 1 cm :
1500 m, one centimeter on the map represents 1500 m in actual distance.
Remember!
A proportion may compare measurements that have different units.
CHAPTER 6.3
The scale on the map of a city is 1/4 inch equals 2
miles. On the map, the width of the city at its widest
point is 3 3/4 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?
The scale on the map of a city is 1/4 inch equals 2
miles. On the map, the width of the city at its widest
point is 3 3/4 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?
¼ inches =
2 miles x = 30 miles, town at widest pt.
3 ¾ inches
x miles
? Time to complete race = 30 miles * 1 hour = 3 hours
10 miles
ANSWER: 3 hours
A 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?
A 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?
3 cm = 125 mi = 62.5 mi, distance bewt landmarks
1.5cm
x
62.5 mi x 1 hr = 1.25 hours or 1 hour 15 minutes
50 mi
Scale Problems
On a Wisconsin road map, Kristin measured a distance of 11 in. from
Madison to Wausau. The scale of this map is 1inch:13 miles. What is the
actual distance between Madison and Wausau to the nearest mile?
Example 2 Continued
To find the actual distance x write a proportion comparing the map distance to the
actual distance.
Cross Products Prop.
x  145
Simplify.
The actual distance is 145 miles, to the nearest mile.
Check It Out! Example 2
Find the actual distance between City
Hall and El Centro College.
Check It Out! Example 2 Continued
To find the actual distance x write a proportion comparing the map distance to the
actual distance.
1x = 3(300)
x  900
Cross Products Prop.
Simplify.
The actual distance is 900 meters, or 0.9 km.
INDIRECT MEASUREMENT Josh wanted to measure
the height of the Sears Tower in Chicago. He used a
12-foot light pole and measured its shadow at 1 P.M. The
length of the shadow was 2 feet. Then he measured the
length of the Sears Tower’s shadow and it was 242 feet
at that time. What is the height of the Sears Tower?
**In a shadow problem, we assume that
a right triangle is formed by the sun’s
ray from the top of the object to the
end of the shadow.
Assuming that the sun’s rays form similar triangles, the
following proportion can be written.
Now substitute the known values and let x be the height
of the Sears Tower.
Substitution
Cross products
Simplify.
Divide each side by 2.
Answer: The Sears Tower is 1452 feet tall.
Nina was curious about the
height of the Eiffel Tower.
She used a 1.2 meter model
of the tower and measured
its shadow at 2 p.m. The
length of the shadow was
0.9m. Then, she measured
the Eiffel Tower’s shadow,
and it was 240 meters.
What is the height of the
Eiffel Tower?
Model height = tower height
Model shadow tower shadow
1.2 m = tower height
0.9 m
240 m
(1.2m)(240m) = tower height
0.9m
333.3 m = tower height
Postulate 6.1
Angle-Angle (AA) Similarity
• AA Similarity- If two
angles of one triangle
are congruent to two
angles of another
triangle, then the
triangles are similar.
Ex: ∠P ≅ ∠T and , ∠Q ≅ ∠S
so △PQR ∿△TSU
There are several ways to prove certain triangles are
similar. The following postulate, as well as the SSS
and SAS Similarity Theorems, will be used in proofs
just as SSS, SAS, ASA, HL, and AAS were used to
prove triangles congruent.
Theorem 6.1
Side-Side-Side (SSS) Similarity
• SSS Similarity- If the
measures of the
corresponding sides of
two triangles are
proportional, then the
triangles are similar.
• EX: PQ = QR = RP
ST SU UT
• So ∆PQR ~ ∆TSU
Theorem 6.2
Side-Angle-Side (SAS) Similarity
• SAS Similarity – if the
measures of two sides of a
triangle are proportional to
the measures of two
corresponding sides of
another triangle AND the
included angles are congruent
then the triangles are similar.
• EX: PQ = QR and <Q <S
ST SU
So ∆PQR ~ ∆ TSU
Example 1: Using the AA Similarity Postulate
Explain why the triangles
are similar and write a
similarity statement.
Since
, B  E by the Alternate Interior
Angles Theorem. Also, A  D by the Right Angle
Congruence Theorem. Therefore ∆ABC ~ ∆DEC by
AA~.
Check It Out! Example 1
Explain why the triangles
are similar and write a
similarity statement.
By the Triangle Sum Theorem, mC = 47°, so C 
F. B  E by the Right Angle Congruence Theorem.
Therefore, ∆ABC ~ ∆DEF by AA ~.
Example 2A: Verifying Triangle Similarity
Verify that the triangles are similar.
∆PQR and ∆STU
Therefore ∆PQR ~ ∆STU by SSS ~.
Example 2B: Verifying Triangle Similarity
Verify that the triangles are similar.
∆DEF and ∆HJK
D  H by the Definition of Congruent Angles.
Therefore ∆DEF ~ ∆HJK by SAS ~.
Ex 1: In the figure, FG≅EG, BE= 15, CF = 20, AE = 9 and DF
= 12. Determine which triangles in the Figure are similar.
Ex 1: In the figure, FG≅EG, BE= 15, CF = 20, AE = 9 and DF
= 12. Determine which triangles in the Figure are similar.
1.
Mark the triangles.
2. When two triangles overlap,
redraw them separately so that the
corresponding parts are in the same
position on the paper.
-mark the triangles
-
write the corresponding angles
and sides.
-
What is the scale factor?
Example 2 pg 300
Find AE and DE.
Example 2 pg 300
AB ║ CD, <BAE ≅ <CDE
and <ABE ≅ <DCE
because they are the
alternate interior angles.
By AA similarity, ∆ABE ~
∆DCE
AB = AE
DC DE
Warm Up
Solve each proportion.
1.
2.
3.
z = ±10
x=8
4. If ∆QRS ~ ∆XYZ, identify the pairs of congruent
angles and write 3 proportions using pairs of
corresponding sides.
Q  X; R  Y; S  Z;
QR = RS = QS
XY YZ XZ
INDIRECT MEASUREMENT
On her trip along the East coast,
Jennie stops to look at the tallest
lighthouse in the U.S. located at
Cape Hatteras, North Carolina.
At that particular time of day,
Jennie measures her shadow
to be 1 feet 6 inches in length
and the length of the shadow
of the lighthouse to be 53 feet 6 inches. Jennie
knows that her height is 5 feet 6 inches. What is
the height of the Cape Hatteras lighthouse to the
nearest foot?
INDIRECT MEASUREMENT
On her trip along the East coast,
Jennie stops to look at the tallest
lighthouse in the U.S. located at
Cape Hatteras, North Carolina.
At that particular time of day,
Jennie measures her shadow
to be 1 feet 6 inches in length
and the length of the shadow
of the lighthouse to be 53 feet 6 inches. Jennie knows that her
height is 5 feet 6 inches. What is the height of the Cape Hatteras
lighthouse to the nearest foot?
Jennie shadow 1ft 6 in = 1.5 ft
Jennie’s height 5ft 6in = 5.5 ft
Light house shadow 53 ft 6 in = 53.5 ft
Jennie shadow = light house shadow  1.5 ft = 53.5 ft  1.5X = (53.5)(5.5)
Jennie height
light house height
X = 196 ft
Answer: 196 ft
5.5 ft
X
Check It Out! Example 2
Verify that ∆TXU ~ ∆VXW.
TXU  VXW by the
Vertical Angles Theorem.
Therefore ∆TXU ~ ∆VXW by SAS ~.
Example 3: Finding Lengths in Similar Triangles
Explain why ∆ABE ~ ∆ACD, and
then find CD.
Step 1 Prove triangles are similar.
A  A by Reflexive Property of , and B  C
since they are both right angles.
Therefore ∆ABE ~ ∆ACD by AA ~.
Example 3 Continued
Step 2 Find CD.
Corr. sides are proportional.
Seg. Add. Postulate.
x(9) = 5(3 + 9)
9x = 60
Substitute x for CD, 5 for BE,
3 for CB, and 9 for BA.
Cross Products Prop.
Simplify.
Divide both sides by 9.
Check It Out! Example 3
Explain why ∆RSV ~ ∆RTU
and then find RT.
Step 1 Prove triangles are similar.
It is given that S  T.
R  R by Reflexive Property of .
Therefore ∆RSV ~ ∆RTU by AA ~.
Check It Out! Example 3 Continued
Step 2 Find RT.
Corr. sides are proportional.
Substitute RS for 10, 12 for
TU, 8 for SV.
RT(8) = 10(12) Cross Products Prop.
8RT = 120
RT = 15
Simplify.
Divide both sides by 8.
Example 5: Engineering Application
The photo shows a gable roof. AC || FG.
∆ABC ~ ∆FBG. Find BA to the nearest tenth
of a foot.
BF  4.6 ft.
BA = BF + FA
 6.3 + 17
 23.3 ft
Therefore, BA = 23.3 ft.
Check It Out! Example 5
What if…? If AB = 4x, AC = 5x, and BF = 4, find FG.
Corr. sides are proportional.
Substitute given quantities.
4x(FG) = 4(5x) Cross Prod. Prop.
FG = 5
Simplify.
Lesson Quiz
1. Explain why the triangles are
similar and write a similarity
statement.
2. Explain why the triangles are
similar, then find BE and CD.
Lesson Quiz
1. By the Isosc. ∆ Thm., A  C, so by the def.
of , mC = mA. Thus mC = 70° by subst.
By the ∆ Sum Thm., mB = 40°. Apply the
Isosc. ∆ Thm. and the ∆ Sum Thm. to ∆PQR.
mR = mP = 70°. So by the def. of , A  P,
and C  R. Therefore ∆ABC ~ ∆PQR by AA ~.
2. A  A by the Reflex. Prop. of . Since BE ||
CD, ABE  ACD by the Corr. s Post.
Therefore ∆ABE ~ ∆ACD by AA ~. BE = 4 and
CD = 10.
Example 1: Measurement Application
Tyler wants to find the height of a
telephone pole. He measured the pole’s
shadow and his own shadow and then
made a diagram. What is the height h of the
pole?
Example 1 Continued
Step 1 Convert the measurements to inches.
AB = 7 ft 8 in. = (7  12) in. + 8 in. = 92 in.
BC = 5 ft 9 in. = (5  12) in. + 9 in. = 69 in.
FG = 38 ft 4 in. = (38  12) in. + 4 in. = 460 in.
Step 2 Find similar triangles.
Because the sun’s rays are parallel, A  F. Therefore ∆ABC ~ ∆FGH
by AA ~.
Step 3 Find h.
Corr. sides are proportional.
Substitute 69 for BC, h for GH, 92 for AB, and 460
for FG.
92h = 69  460
h = 345
Cross Products Prop.
The height h of the pole is 345 inches, or 28 feet 9 inches.
Check It Out! Example 1
A student who is 5 ft 6 in. tall measured
shadows to find the height LM of a flagpole.
What is LM?
Step 1 Convert the measurements to inches.
GH = 5 ft 6 in. = (5  12) in. + 6 in. = 66 in.
JH = 5 ft = (5  12) in. = 60 in.
NM = 14 ft 2 in. = (14  12) in. + 2 in. = 170 in.
Step 2 Find similar triangles.
Because the sun’s rays are parallel, L  G. Therefore ∆JGH ~ ∆NLM
by AA ~.
Step 3 Find h.
Corr. sides are proportional.
Substitute 66 for BC, h for LM, 60 for JH, and 170
for MN.
60(h) = 66  170
h = 187
Cross Products Prop.
Divide both sides by 60.
The height of the flagpole is 187 in., or 15 ft. 7 in.
Example 3: Making a Scale Drawing
Lady Liberty holds a tablet in her left hand. The tablet is 7.19 m long and
4.14 m wide. If you made a scale drawing using the scale 1 cm:0.75 m,
what would be the dimensions to the nearest tenth?
Example 3 Continued
Set up proportions to find the length l and width w of the scale drawing.
0.75w = 4.14
w  5.5 cm
9.6 cm
5.5 cm
Check It Out! Example 3
The rectangular central chamber of the Lincoln Memorial is 74 ft long and
60 ft wide. Make a scale drawing of the floor of the chamber using a scale
of 1 in.:20 ft.
Check It Out! Example 3 Continued
Set up proportions to find the length l and width w of the scale drawing.
20w = 60
w = 3 in
3.7 in.
3 in.
Example 4: Using Ratios to Find Perimeters and Areas
Given that ∆LMN:∆QRT, find the perimeter P and
area A of ∆QRS.
The similarity ratio of ∆LMN to ∆QRS is
By the Proportional Perimeters and Areas Theorem, the ratio of the triangles’
perimeters is also
, and the ratio of the triangles’ areas is
Example 4 Continued
Perimeter
13P = 36(9.1)
P = 25.2
Area
132A = (9.1)2(60)
A = 29.4 cm2
The perimeter of ∆QRS is 25.2 cm, and the area is 29.4 cm2.
Check It Out! Example 4
∆ABC ~ ∆DEF, BC = 4 mm, and EF = 12 mm. If P = 42 mm and A = 96
mm2 for ∆DEF, find the perimeter and area of ∆ABC.
Perimeter
12P = 42(4)
Area
122A = (4)2(96)
P = 14 mm
The perimeter of ∆ABC is 14 mm, and the area is 10.7 mm2.
Lesson Quiz: Part I
1. Maria is 4 ft 2 in. tall. To find the height of
a flagpole, she measured her shadow and
the pole’s shadow. What is the height h of
the flagpole?
25 ft
2. A blueprint for Latisha’s bedroom uses a
scale of 1 in.:4 ft. Her bedroom on the
blueprint is 3 in. long. How long is the
actual room?
12 ft
Lesson Quiz: Part II
3. ∆ABC ~ ∆DEF. Find the perimeter and area of ∆ABC.
P = 27 in., A = 31.5 in2
Ch 6.3 Activity pg 298
• Draw ⍍DEF with m∠D = 35, m∠ F = 80, and DF = 4
cm
• Draw ⍍RST with m∠R = 35, m∠ S = 80, and ST = 7
cm
• Measure EF, ED, RS, RT
• Calculate the ratios FD, EF, and ED
ST RS
RT
1. What can you conclude about all of the ratios?
2. What are the minimum requirements for two
triangles to be similar?
Example 4: Writing Proofs with Similar Triangles
Given: 3UT = 5RT and 3VT = 5ST
Prove: ∆UVT ~ ∆RST
Example 4 Continued
Statements
Reasons
1. 3UT = 5RT
1. Given
2.
2. Divide both sides by 3RT.
3. 3VT = 5ST
3. Given.
4.
4. Divide both sides by3ST.
5. RTS  VTU
5. Vert. s Thm.
6. ∆UVT ~ ∆RST
6. SAS ~
Steps 2, 4, 5
Check It Out! Example 4
Given: M is the midpoint of JK. N is the
midpoint of KL, and P is the midpoint of JL.
Check It Out! Example 4 Continued
Statements
Reasons
1. M is the mdpt. of JK,
N is the mdpt. of KL,
and P is the mdpt. of JL.
1. Given
2.
2. ∆ Midsegs. Thm
3.
3. Div. Prop. of =.
4. ∆JKL ~ ∆NPM
4. SSS ~ Step 3