a.
Ch. 6 Getting Started
AWM1O
Assignment
PART A: How to Cross Multiply
When we need to solve for a variable in a proportion, we have to cross multiply in order to solve for it.
Variable: an unknown value represented by a letter.
Steps to solve for a variable in a proportion:
1) Multiply the variable with the number diagonal from it.
2) Multiply the other 2 numbers that are diagonal from each other.
3) Divide both sides by the number in front of the variable (multiplied to it).
Examples:
7
10
735
s_8
310
‘01
Old
92
eea—(6
K,
— C—
0!
1
S 01
9
r7
rIb
‘S
6
2
01*
96
:uopsodoj ip aAqo
PART B: How to simplify fractions and to put them into mixed numerals.
Mixed numeral: a whole number and a fraction together.
Greatest common factor: largest number that will divide evenly into 2 (or more) numbers.
Multiplying fractions: multiply the numerators and then multiply the denominators. Remember that a
whole number has a 1 as a denominator.
Examples:
1)i2÷6=
18÷6=3
4)27—9=3
36 ÷9 =4
2)÷8=
56÷8=7
3) j2÷6=
42÷6=7
)
3
/s
S
1
8
=
4
—
32
20 ÷4=5
6)+5914/5
20÷5=5
StnpN1 each. Vrte your aus\ver a a inked number when posib1e.
.6
i).
12
32
3)
.45
2)
4)
9
—.
12
3O
4
I2
320
)45
9)
36
96
.—
11)
fl)
30
:72
l0
12)
16)
60
96
1?)
—
19)
:.
‘0
32
126
•6
42
8
15)
—
..
18)
20)
20
36
36
45
Some more practice...
Simplify co’h. Write your answer a a mixed number when possible,
9)
iS
42
30
24
42
:
1O
24
Find eaeb product.
IL)}.9
13) 7!
3
14) 2
15) 10
16)
17) 2x’
U)
2
19) 2
l
.
j
20)
Name:
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Ch. 6.1 Similar Figures
Notes
When two figures or shapes are similar, one figure or shape has dimensions (length and width) that are
proportional to the other figure. We will look at many different types of figures and determine whether
or not they are similar.
Similar Figures:
Congruent:
Corresponding Angles:
Corresponding Sides:
Example 1
Debbie works at an art gallery and is adjusting a bear image to fit on a certain wall. She is not sure if she
changed the length and width by the same amount and wants to find out if she has used the right ratio for
each dimension.
a) What is the ratio of lengths (the horizontal measurement)?
b) What is the ratio of widths (the vertical measurement)?
c) Has the same ratio been used in both cases?
d) Are the two pictures similar?
Example 2
Steve and Tanya are moving from their home in Chilliwack to a new larger home in Hope. Tanya loves
her garden so much she wants to build a bigger one in Hope but keep it exactly the same shape. Steve
drew out the plans for the new larger garden on the right.
2O.4rn
a) Are these two gardens similar?
Example 3
A tissue company creates rectangular tissue with a length of 9 cm and a width of 10 cm. They want to
increase the size of their tissue for people with bigger noses by a factor of 1.7. What would be the new
dimensions of the larger tissue?
Let’s say the tissue company wanted to decrease the size by 2 for people with smaller noses. What would
be the dimension of the smaller tissue?
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Ch. 6.1 Worksheet#1
Assignment
Each pair of figures is similar Find the missing side.
1)
2)
?2A
5)
6)
A
8)
73
1aj
10)
x
7.8
62.4
11. In the following pictures below write down all of the corresponding sides and corresponding angles
between the two shapes.
w
AB=
BC=
<B=______
CD=
B
DA=
‘C
12. Which property proves that zABC is
similar to ADEF?
A.
ARAC
DF
DE
—
a
AC
C
L4=LD
a
AC
iC
=
-
—
ED
DE
EP
A
I cm
B
F
lcni
E
13. If zRST is similar to zLMN and angle measures of ALMN are as follows,
what are the angle measures of zSRST?
LN=17°
M=78°
ZL=85°
T
MA
14. Find the lengths of the smaller figure.
A.
21cm
F
LP=
I
op=
p
NO=
MN=_____
15. Staceyis bored and draws a scale drawing of her room. If the actual size of her longest wallA is
12.75’, what is the size of all of her other walls in real life. Use corresponding sides to solve for each wall.
WallA: 12.75’
Wall
r=ar
=
B:
Wall C:
Wall D:
Wall E:
Wall F:
Name:
AWM1O
Ch. 6.1 Worksheet#2
Assignment
Answer each question and roun d your answer to the nearest Whole number
.11) Santa Cniz and Johnstown are 84 ml from
cities
eai± other. How far apart would
i4mi?
in;
thasasealeofl
beonamaptha
12) Amaphasascaleoflin: l6mi. IfSaata
Cniz. and Brisbane are 64 iiii apart then they
are bow far apart on the nap?
1.3) A.rnaphasascaieoflcm:i2km. If
Greenwood and Rivertown are 9 cm apart
an the map then how far apart are the real
cities?
14) A model house is 6 in tall If it was built
wathascaleofirn 3fttbenhowtaflasthe
acal house?
15) A particular giraffe is 16 taI1 A model of
itwasbuiltwithascaleoflm 811 How
tall is the model?
16) Apartoulartrairis I2fttalL Amodeiafit.
wasbuiltwithascaleoflin:4ft. Howtall
model?
is
17) A6fttallmanstandingnexttoaladdcr
casts a 3 ft shadow If the ladder casts a
shadow that is? ft long then how tall is it9
18) ifa3fttallcarcastsa9ftiongshadowthen
how tall isa woman that casts a 18 ft
shaw?
19) Astatuethatis 15fttallcastsasha*:lawthat
is 5 ft long. Find the length oi the shadow
thataI8tIaduitgiraflecasts.
20) lfa8fttallbabygiraffecastsa4ftlong
shadow then how long is the shadow that a
1fttaIl statue casts?
21) Franklin and Clayton are SC) km from each
other, how far apart: would the cities be Cfl
amapthathasascaleofl cm: 10km?
22) Find the thstance between Rwertown and
Yorkshire on a map with a scale oft cm: S
kmifthey areactuaflyl2kmapart
23) Rivertown and Abbots Rise are 2 in apart on.
aniapthathasascaleofl in :lmi, How
far apart are the real cities?
24) Find the distance between Kumba and
Midwayiftheyare5inapartonamapwith.
a scale of I in: 5 mi.
25) A model motorcycle has a scale of I in : S ft.
If the real motorcycle is lOft long then how
long is the model motorcycle?
26. Two triangles are similar. One has sides of 8 m, 5 m, and 6 m. If the longest side of the second triangle is 5
m, what are the lengths of the other two sides?
27. A pentagon has interior angles of 108°, 204°, 630, 120°, and 45°. Rudy wants to draw a similar pentagon
with sides twice as long as the original. What size will the angles be?
28. Marie-Claude has a series of four nested funnels in her kitchen that are similar to the one shown in the
diagram. If the other three funnels have top diameters of 10cm, 8 cm, and 6 cm, find the measures of the
remaining parts for all three funnels.
a)
Funnel
Height:
1.
16cm
2.
/
3.
10cm
b) Stem Height:
1.
2 cm
2.
3.
Name:
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Ch. 6.2 Determining if Two Polygons are Similar
Notes
When we would like to determine if two shapes are similar we must check that their
If these two things are not equal then the shape will not be
are the same as well as their
similar.
Review: Try to solve these in your head
30
x
10100
72108
x12
5045
9
x
—
Scale Factor:
Example 1
Are the following shapes similar? If so prove it
55cm
:::r
27.5cm
9
20,5cm
41cm
Rcacdos Posier
Original Posior
What is the scale factor of Ricardo’s poster?
Example 2
To the right are some cedar hats Aithea is designing in three different sizes
a) Are they the same shape?
J1\
45°\
/45
Small
b) Are they similar figures?
Medium
Large
Example 3
Lionel Richie is building a model boat to “sail on down the line.” A model was constructed before the
actual build. The model is on the left and the real size is on the right. Are they similar? What is the scale
factor?
36”
N
//15
15
“p
25”
‘___/‘25”
T
0”
S
4”
Example 4
Rhoniel is an interior decorator who is creating a wall pattern with similar parallelogram stencils. She
created the three similar parallelograms shown. She wants to make the corresponding sides of each
stencil have the same four colors: brown, yellow, blue, and orange.
a) Help Rhoniel list the pairs of corresponding
C
B
//
//
sides to figure out her paint colors.
/
/
//
/115
\
8 cm/’
cm
F
/1
/ 12 H
\ //
/
A
/
/‘D
°
b) What scale factor was used on ABCD to
create EFGH?
K
I
izs°
/
‘10cm
16cm
c) What is the measure of side BC?
d) What is the measure of angle D?
e) What is the angle measure of <L? How do you know? Explain your Answer
M
cm
Name:
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Ch. 6.2 Worksheet
Assignment
1. Pierre drew two regular hexagons (6-sided figures with all sides equal in length). Are the two hexagons
similar? Why or why not?
2. Frank enlarges a photo to poster size. The original photo is 4 inches by 6 inches. If Frank enlarges it to
1 m by 1.5 m, will it be similar in shape to the original? (Hint: Draw a diagram)
3. Are the following shapes similar? Show your work to support your answer.
A
D
z
B
4. One cylinder has a radius of 25 cm and a height of 35 cm. Another cylinder has a radius of 30 cm and a
height of 40 cm. Are the cylinders similar? Show your calculations. (fill in the diagram)
5. 2 cm on a map refers to 60 km in real life. If Chilliwack to Whistler is 191 km, how far would that be on
the map?
6. The scale on a map is 2.5 cm:500 m.
a) What distance is represented by a 12.5-cm segment on the map?
b) How long would a segment on the map be if it represented 1.5 km?
7. Cohn states that the following two figures are similar, but Tai disagrees, saying that they don’t have
enough information. Who is right? Show your calculations.
lOin
5cm
IBm
9 cm
5cm
8. While he was at the pet food store, Jeremy saw three different sized dog mats. They measured 36
inches by 28 inches, 27 inches by 21 inches, and 24 inches by 18 inches. Are all the mats similar? Show
your calculations.
Name:
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Ch. 6.1-6.2 Worksheet
Assignment
****please draw diagrams with all of your work****
1. On a blueprint, a room measures 2.75 inches by 1.5 inches. If 1 inch represents 8 feet, what will be the
dimensions of the room?
2. Jason wants to build a model of his house. He will build the model using a scale where 5 cm represents
2 m. If one room is 6.5 m long, 4.8 m wide, and 2.8 m tall, what will its dimensions be in the model?
3. Redo Question 2 by first determining a unit scale (the number of centimeters that represent 1 meter),
then calculating length, width, and height for the model.
4. If a house is 40 feet long, 35 feet wide, and the top of the roof is 27 feet above ground level, what will
the corresponding dimensions be of a model built so that 1 foot is represented by 12 inch?
5. Theresa folds origami paper to make stacked boxes. The outer box is 12 cm by 8 cm by 4 cm. Theresa
would like to make two smaller, similar boxes, each scaled down by 5% of the previous box. What are
the dimensions of the two smaller boxes?
6. Are the two pentagons shown below similar? If so, explain how you know. If not, explain what you
would need to know. (Angles marked with the same symbol are equal.)
.
)
6cm
Li
7. The scale on a map is 2.5 cm: 500 m.
a) What distance is represented by a 12.5-cm segment on the map?
b) How long would a segment on the map be if it represented 1.5 km?
8. Show whether a rectangular prism that is 6m X 10 m X 8m is similar to one that is 4m X 7m X Sm.
9. While he was at the pet food store, Jeremy saw three different sized dog mats. They measured 36 inches
by 28 inches, 27 inches by 21 inches, and 24 inches by 18 inches. Are all the mats similar? Show your
calculations.
10. Gina drew a scale representation of a field (see Polygon C). For instance, she drew a line
2.2 cm to represent the actual measure (20 m) of this side (AD). Gina needs to complete the
drawing of Polygon D so that both polygons are similar.
Pulygon D
Poiygcii C
H
D
1
;_J!i__
3.’
/
28m
in /
in
/1
-————
c
F
11. The two tents below are similar. What is the height of the smaller tent.
G
12. A small soccer field is to be enlarged, though its shape will stay the same. What will be the area
of the new field?
I
I
S
I
I
S
S
S
I
I
I
5
2idi
I
(O :J
13. The scale on the neighborhood map on the next page shows that 1 cm on the map represents an
actual
distance of
2.5 km.
a. On the map, Waltham Street has a length of 14 cm. Using the scale, what would be the length of the
actual street?
b. Matapan Street has an actual length of 25 km. Show your work to find the right length of the Street Ofl
the map. Then use a ruler to see if your calculation is correct.
Name:
Ch. 6.3 Drawing Similar Polygons
AWMIO
Notes
Sometimes when we are enlarging a figure we can look to the scale factor to let us know how much larger
or smaller our new object will be drawn.
Scale Factor:
Scale Factor greater than 1 will
Scale Factor that is a fraction will
Example 1
If you were to enlarge the figure below by a factor of 1.5, what would be the dimensions of the larger
version be. Include the side lengths and angle measurements.
4cm
3.5cm
cm
Example 2
Lauren illustrates “how-to” manuals that show customers how to assemble furniture. One of her co
workers went home sick, and she was given the following diagram of a triangular shelf and told to
redesign it. The triangular face of the new shelf has one side length of 60 cm and is defined as a similar
triangle.
Now Lauren has to figure out the dimensions of the rest of the triangle. She needs to figure out what scale
factor her co-worker used. Is there more than one triangle possible?
Example 3
Determine what scale factor was used to create the larger piece and use the scale factor to calculate the
missing side lengths.
Example 4
Use the ratio method to create a pentagon that has been scaled down by a factor of ½.
How could you make a shape that is 1/3 bigger than the one below.
Name:
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Ch. 63 Drawing Similar Polygons
Notes
Sometimes when we are enlarging a figure we can look to the scale factor to let us know how much larger
or smaller our new object will be drawn.
Scale Factor:
Scale Factor greater than 1 will:
Scale Factor that is a fraction (less than 1) will:
Example 1
If you were to enlarge the figure below by a factor of 1.5, what would be the dimensions of the larger
version be. Include the side lengths and angle measurements.
Example 2
Lauren illustrates “how-to” manuals that show customers how to assemble furniture. One of her co
workers went home sick, and she was given the following diagram of a triangular shelf and told to
redesign it. The triangular face of the new shelf has one side length of 60 cm and is defined as a similar
triangle.
Now Lauren has to figure out the dimensions of the rest of the triangle. She needs to figure out what scale
factor her co-worker used. Is there more than one triangle possible?
Example 3
Determine what scale factor was used to create the
larger piece and use the scale factor to calculate the
missing side lengths.
a
6 in
61n
4
B in
Example 4
Use the ratio method to create a pentagon that has been scaled down
by a factor of ½.
How could you make a shape that is 1/3 bigger than this one.
b
Name:
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Ch. 6.3 Drawing Similar Polygons
Assignment
1. Use graph paper to construct a figure similar to the one given, with sides that are 1/2 the length of the
original. Explain how you know that the corresponding angles are equal.
H
L___
2. Draw and label the lengths of the sides of a rectangle that has a length of 8 cm and is similar to a
rectangle that has a width of 10 cm and a length of 20 cm.
3. Barbie’s house is 55 ft wide and 40 ft deep. A drawing of her property shows the house is 10 in wide
and 7.3 in deep. What scale was used on the drawing?
4. Xavier is building a staircase using scale drawings. On the drawing, the height of one stair is 0.5 cm and
the depth is 0.9 cm. Xavier will use a scale factor of 40 to build the stairs. Calculate the height and depth
of the stairs he will build.
5. The scale ratio (model: original) between two diagrams is 3:5. If one measure on the model is 45 mm,
what was the measure on the original?
6. Simian has built two end tables. The second table is a slightly larger version of the first. Given the
dimensions below, calculate what scale factor Simian used to make the larger table.
7. Find the missing angles and measures of the following diagram.
I 1ff
4Scrn
3cm
42ciri
r
8. A craft store uses small gift boxes to wrap purchases. They have one box that is 20 cm by 12 cm by 5
cm. Another box is larger by a scale factor of 1.3. What are the dimensions of the larger box?
9. Draw a rectangular prism similar to the one shown below with sides that are 1/2 the length of the
original.
i
cm
cm
10. Hazuki made a kite with the dimensions shown below. She decided it would work better if it were
bigger. If her new kite tail has a length of 49 cm, what scale factor did she use, and what are the kite’s
other dimensions?
11. A sporting goods store has miniature versions of tents on display. A six-person tent is 12’longby
10’wide. The miniature version has a length of 1 ½ inches.
a) What is the width of the miniature version?
b) What is the scale ratio (miniature:actual)?
Ch. 6.4 Similar Triangles
AWM1O
Notes
Section 6.2 defined that two figures are similar if their corresponding sides are proportional to each other and
their corresponding angles are the same. The same rules apply to triangles, but you can determine if they are
similar by using even less information.
Two triangles are similar if one of the two following requirements is
true.
• Any two of the three corresponding angles are equal
• One pair of corresponding angles is congruent and the
corresponding sides adjacent to these angles are proportional.
, -1 S
I\
I\
i3ft
I
259
\
Rft
Example 1
Are any of the following triangles similar?
a
/840
Example 2
Sven is designing a T-shirt and wants to use several triangles in her design. She drew triangle ABC below to
represent the triangular shape she wants to use in her design. The side lengths of the triangle are as follows:
AB
4
BC
5
AC
6
DC
2.5.
and
EC=3
B
D
Find the length of ED.
Example 3
Roberto and Marcos tie wires to either side of an artificial tree as
part of the set-up of a concert stage. They decide to attach the
wires so that they both make a
angle within a right triangle.
350
Are the right triangles created by the wires similar triangles?
Example 4
Given that MBC in similar to LRST, AB is 6 cm long, BC is 5 cm long, and RS is 8 cm long, find the length of a
second side in ARST. Can you find the length of the third side? Explain your answer.
Example 5
Ravi notices that a 2-rn pole casts a shadow of 5 m, and a second pole casts a shadow of 9.4 m. How tall is the
second pole? (sketch the situation)
Name:
Ch. 6.4 Worksheet
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Assignment
1. Given the two triangles below, find the length of ii.
C
b
aSi
N
b) Are these triangles similar?
c7n
2. In each of the diagrams below, LABCis similar to LXYZ Find the length of the indicated side (to one
decimal place).
cC
6.1
I
B
C
8
3. Gladis thinks that any two isosceles triangles will be similar. Use examples to prove or disprove her
belief.
An isosceles triangle
has two sides equal
in length, and two
angles of equal
measure.
4. Ethel notices that a 4-rn pole casts a shadow of 8 m, and a second pole casts a shadow of 22 m. How tall
is the second pole? (please provide a diagram)
5. Midge has cut out two triangular shapes from a block of wood, as shown below. She says that the two
shapes are similar. Is she correct? Show your calculations.
2 I
6. Julian is visiting the Manitoba Legislative Building in
Winnipeg, where he sees the statue of Louis Riel. Use the
information in the diagram to find the height of the statue.
Round your answer to the nearest whole number.
Sfta
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Ch. 6 Review
Assignment
1. On a blueprint, a room measures 2.75 inches by 1.5 inches. If 1 inch represents 8 feet, what will be the
dimensions of the room?
2. Jason wants to build a model of his house. He will build the model using a scale where 5 cm represents
2 m. If one room is 6.5 m long, 4.8 m wide, and 2.8 m tall, what will its dimensions be in the model?
3. Redo Question 2 by first determining a unit scale (the number of centimeters that represent 1 meter),
then calculating length, width, and height for the model.
4. Gina drew a scale representation of a field (see Polygon C). For instance, she drew a line to represent
the actual measure (20 m) of this side (AD). Gina needs to complete the
drawing of Polygon D so that both polygons
PuIgou (.
Pohoii D
are similar.
2!
A
/7
/
1
D
28m
/
1
33rn
B33m
G
F
5. Find the missing angles and measures of the following diagram.
4.4 cn
4.5cm
3an
1
42cm
6. Draw a rectangular prism similar to the one shown below with sides that are 1/2 the length of the
original.
cm
7. The scale on a map is 2 cm: 600 m:.
a) What distance is represented by a 16-cm segment on the map?
b) How long would a segment on the map be if it represented 22 krn?
8. Zora says that the two rectangles below are not similar because 60/50 does not equal 100/30. Is Zora
right? Explain.
A
60cm
100cm
30cm
0
C
9. The lengths of the sides of a pentagon are 2”, 6”, 10”, 14”, and 24”. Calculate the lengths of the sides of a
similar pentagon if the shortest side is 5”. Draw a diagram
10. If a man casts a shadow that is 3.8 m long at the same time that an 8-rn flagpole casts a shadow that is
15 rn long, how tall is the man? (draw a diagram)
Name:
Block:_______
Date:
Ch. 6 Practice Test
AWM1O
Test
(30 marks)
1. In order to show two figures are similar you must prove...?
2. The scale on a map is lcm:3km, how many cm on the map would 36 km be?
3. A picture of Carla’s pen in a magazine is 21 cm long, the real pen is 7 cm, what is the
scale factor used to create the picture?
4. In the picture below, what is the scale factor used to create Ricardo’s poster?
a) 2
55cm
27.5cm
b)½
20.5cmL
41cm
c) 3/4
RicardWs Poster
d) 4
Original Poster
5. Picture two similar figures. One of the figures is 2 times the size of the other. If you
create a similar shape that is 2 times as big, the angles will be...
a) doubled
c) the same
b) tripled
d) halved
6. Solve the following:
3
5
—
x
—
35
4cm
7. What would the dimensions of this picture be if it
was enlarged by a scale factor of 1.75?
8. In the diagram, which side corresponds to
side DC?
a)AD
b) HG
c) FG
d) BC
13Gm
AB
tOrn
7.2 m
E-.
-
9. Determine what scale factor was
used to create the larger piece.
b
10. Which statement correctly expresses a similarity between two triangles in the
diagram below?
E
a) AAGB-iCED
F
b) MGB —MCF
c) MCFADE
d) MFC —tIADE
C
-
----
-
1-
1
B
D
C
11. Which of the right triangles below are similar?
Hi
a) I and II only
b) II and IV only
c) I, II and III only
d) II, III and IV only
Iv
z
45 m
12. Which property proves that zXABC is similar to tXDEF?
A
AB_AC
DEDF
B. AC=ED
C.
L4LD
D
AC DE
BCEF
2 cm
E
7
V
/
1 cm
D
/
-
4 cm
D
/ •) cm
13. A small soccer field is to be enlarged, though its shape will
stay the same. What will be the area of the new field?
2 yd
60yd
I
I—
—
Part B—short answer section;please show all your workfor full marks
1. The lengths of the sides of a pentagon (5 sided shape) are 2”, 6”, 10”, 14”, and 24”. Use scale
factors to calculate the lengths of the sides of a similar pentagon if the shortest side is 5”. (2
marks)
2. Given that the two figures shown are similar, determine the values ofxandy (4 marks)
8.5m
2EEJ
x
3. To determine the distance across a river (distance AB), Lila took the following
measurements. Assuming the two triangles in the diagram are similar, how wide is the
river? (2 marks)
4. If a man casts a shadow that is 3.8 m long at the same time that an 8-rn flagpole casts a
shadow that is 15 m long, how tall is the man? (2 marks)
5. Amin cut out two blocks of wood as indicated. Are the two blocks similar in shape? Round
your final calculations to two decimal places. (2 marks)
in
7 In
275 in
4 in
9.Zin
6. Joanne knitted a blanket that measures 174 cm by 230 cm. Her sister asked Joanne to make
a matching one for her son. If Joanne wants to make a similar blanket using a scale factor of
0.55, what will its dimensions be? Include a diagram (3 marks)
7. In the following diagram, AB is parallel to ED, AB is 8 m, AC is 12 m, and CE is 7 m.
Calculate ED to one decimal place. (2 marks)
7
iii