Name: ___________________________
Period: ___________
8.1 Drawing Dilations by Hand
1. Draw the image of the figure under a dilation with a scale factor of 2, using point A as the center of dilation.
2. Draw the image of the figure under a dilation with a scale factor of 3, using point P as the center of dilation.
3. Draw the image of the figure under a dilation with a scale factor of ½ using point A as the center of dilation.
Need more? Try searching Khan Academy for “Dilations or scaling around a point”
For the dilations in the coordinate plane assume the center of dilation is always the origin. Draw the image under a
dilation using the indicated scale factor.
4. Scale factor: 2
5. Scale factor: 1/2
6. Scale factor: -1
7. The coordinates of the pre-image are:
A(1,-6), B(-4,4) and C(-1,-3)
Using a scale factor of 6.6, what would the coordinates
of the image be?
A __________ B__________ C _________
8. Investigate it: If you were to dilate a line segment would the image be parallel to the pre-image? Try dilating a line
around a point not on the line, or a point on the line. What happens?
Name: __________________________ Similarity Investigation
8.2a AA,SSS, SAS Similarity
Similar Triangle Investigation 1
Is AA (angle-angle) enough to say 2 triangles are similar?
You need rulers, protractors, pencils and graph paper will probably help
Step 1: Draw any triangle ABC
Step 2: Construct a 2nd triangle DEF with ∠𝐷 ≅ ∠𝐴 and ∠𝐸 ≅ ∠𝐵
Question: What must automatically be true about ∠𝐶 and ∠𝐹? _______________________
How do you know? _________________________________________________________
Step 3: Carefully measure the lengths of the sides of both triangles. Compare the ratios of the
corresponding sides. Is
𝐴𝐵
𝐷𝐸
≈
𝐴𝐶
𝐷𝐹
≈
𝐵𝐶
𝐸𝐹
?
Step 4: Compare your results with others at your same table. You should be able to state a conjecture.
AA similarity conjecture:
If ________ angles of one triangle are congruent to __________ angles of another triangle
then _______________________________________________
Drawings:
Similar Triangle Investigation 2
Is SSS Sufficient to say two triangles are similar?
You need rulers, protractors, pencils and graph paper will probably help
Step 1: Draw any triangle ABC.
Step 2: Construct a second triangle, DEF, whose side lengths are a multiple of the original triangle. (Your
second triangle can be larger or smaller) – Recall how to make an SSS construction which can be found in
your notes from last semester. If all else fails, by all means you can use a phone too look up how to make
an SSS triangle.
Step 3: Compare the corresponding angles of the two triangles, and also compare the results of your
peers. Do their findings match yours?
SSS similarity conjecture:
If three sides of one triangle are proportional to the three sides of another triangle then the two triangles
are __________________.
Drawings:
Similar Triangle Investigation 3
Is SAS Sufficient to say two triangles are similar?
You need rulers, protractors, pencils and graph paper will probably help
Step 1: Construct two different triangles that are not similar but have two pairs of sides proportional and
the included angles congruent. (For a hint I made this in geogebra to sort of illustrate how you should
start this process. But it would be totally unoriginal if you copied my work…)
Step 2: Compare the measures of the corresponding sides and angles. Share you results with your peers,
and finish the conjecture.
SAS similarity conjecture –
If the sides of one triangle are proportional to the two sides of another triangle and ___________, then the
______________
Similar Triangle Problems 4
1) 𝑆𝑎𝑙𝑙𝑦 𝑠𝑜𝑙𝑣𝑒𝑑 𝑡ℎ𝑖𝑠 𝑝𝑟𝑜𝑏𝑙𝑒𝑚 𝑎𝑛𝑑 𝑔𝑜𝑡 𝑥 = 24. 𝐷𝑒𝑠𝑐𝑟𝑖𝑏𝑒 ℎ𝑒𝑟 𝑒𝑟𝑟𝑜𝑟:
2) Solve for x
3) In each group, which triangles are similar to triangle A?
Name___________________________________
8.2 Similar Triangles
©x d2Q0s1z5x _KtuCtDaM fSUoMf_tHwnaXrkeO [LyLeCK.V Q kAyltl] jrsiJgohDtIsW Srie^sseKrRvOeddI.
Are the triangles in each pair are similar? If so, state how you know they are similar (AA~,
SAS~, or SSS~) and complete the similarity statement.
1)
L
M
2)
B
A
C
K
H
F
G
KLM ~ ______
F
G
FGH ~ ______
3)
F
G
4)
M
21
12
12
15
L
E
5
K
13
4
M
77
W
L
44
EFG ~ ______
U
44
V
WVU ~ ______
5)
6)
G
F
13
H
50
13
40
P
G
78
U
78
88
77
R
F
36
Q
V
PQR ~ ______
111
W
UVW ~ ______
Worksheet by Kuta Software LLC
-1-
©F b2Q0w1j5q ]Kqupt\ag qSuoQfwtSwcaqrDen VLlLeCg.C R lALlZlv MrGiygWh^tRsg [roeqsVeBrRvtehd\.r u wMxaQd[e_ XwaiNtchl gIKnqfbiMnxiDt]eD VG^eMo]mgeftnrcyi.
Find the missing length. The triangles in each pair are similar.
7)
V
8)
9
T
?
C
70
D
R
63 °
U
8
13
56
E
S
F
63 °
65
?
G
9)
H
10)
S
T
?
?
39 °
T
66
F
S
R
143
R
156
C
60
84
39 °
B
G
D
60
State if the triangles in each pair are similar. If so, state how you know they are similar and
complete the similarity statement.
11)
12)
D
U
30
10
40
E
8
110
D
R
C
33
P
T
121
32
E
Q
CDE ~ ______
PQR ~ ______
Worksheet by Kuta Software LLC
-2-
©Q c2w0]1c5z OKeuZtcaz cSgokfXtVwVasrQeR qLtLvCz.K _ `AYlPlg lraiMgqhetEsW ^rseesyeMrivRePdK.k G ]MpazdOeL `wXi\thhA OIRnffmiTnuiDtyeu XGgeXokmpeItZrdy\.
Name: _________________________________ Period: _________
8.3a Dilation of a Line Segment
1. Given 𝐴𝐵, draw the image of 𝐴𝐵 as a result of the dilation with center at point C and scale
factor equal to 2. Show your construction marks.
D
E
A
B
C
2. Describe the relationship between 𝐴𝐵 and its image.
3. Connect points D and E with a line segment.
4. Dilate 𝐷𝐸 using point C as the center of dilation, with a scale factor of
1
.
2
5. Describe the relationship between 𝐷𝐸 and its image.
6. In general, what happens when you dilate a line segment by a point not on the line?
C
A
T
7) 𝐶𝑇 is dilated from point A by a scale factor of ½.
What are the new coordinates of each point?
C’ _____
A’_____
T’ _____
8) What did you notice about point A?
____________________________________________
9) Describe the relationship between 𝐶𝑇 and its image.
____________________________________________
10) On the graph above, graph a diagonal line segment with endpoints in the first quadrant. Label them
D and G. Do not make a horizontal or vertical line.
Coordinates of D _______________ Coordinates of G ____________
11) Label a point O on 𝐷𝐺 so that O is somewhere between the two endpoints.
Coordinates of O ______________
12) Choose a scale factor _________ (this can be a fraction or a whole number)
13) Dilate 𝐷𝐺 from the point O. Label D’,G’
Coordinates of D’ _____________ Coordinates of G’ ____________
14) Describe the relationship between 𝐷𝐺 and its image. _______________________________________
Explain why you know this to be true:
Name: __________________________
Period: _________
8.3b Dilating Segments
In this investigation you will learn the effect of dilations of line segments and the properties that remain true of the image
of the segment after a dilation.
Part 1: Pick arbitrary points for the end points of your segment. One end has coordinates (𝑥, 𝑦) and let the other end point
will have coordinates(𝑝, 𝑞). Graph these arbitrary points in the graph
below.
Part 2: Using an arbitrary scale factor of k, and using the origin as the
center of dilation, what would be the coordinates of the image of (𝑥, 𝑦)
after a dilation? (Use the rule for dilations learned yesterday)
(𝑥, 𝑦) →_______________
What would the coordinates of (𝑝, 𝑞) be after the same dilation?
(𝑝, 𝑞) →_______________
Graph these points in the plane, recall they should line up with the center of dilation and the pre-image, and maintain the
same ratio.
Part 3: What is the slope of the pre-image segment?
Part 4: What is the slope of the image segment?
Part 5: Use algebra to show that the segments have the same slope. (Hint: you will need to factor)
Part 6: What does your result from part 5 mean? Think about what similar slopes indicate.
Part 7:
a)
b)
c) What is the slope of 𝐴𝐵?
d) What is the slope of 𝐴′𝐵′?
e) What can you conclude about 𝐴𝐵 𝑎𝑛𝑑 𝐴′𝐵′?
Name: ______________________________ Period: ________
8.4 Similarity Transformations
Two plane figures are similar if and only if one can be obtained from the other by a sequence of
translations, reflections, rotations and/or dilations.
Use the definition of similarity in terms of similar transformations to determine whether the two figures
are similar. EXPLAIN your answer on the lines provided, include all transformations needed.
1)
4)
2)
5)
3)
6)
Describe the transformations that would prove the circles are congruent. Include by what vector you
would translate the centers and what scale factor you would use.
7)
8)
Prove: all circles are similar.
Given: Circle with center D and radius j, and Circle with center S and radius c.
Prove: Circle D is similar to Circle S.
A) First transform circle D with the translation along vector ⃗⃗⃗⃗⃗
𝐷𝑆
Under this translation, the image of point D is ________
The center of circle D’ must lie at point _________
B) Now transform circle D’ with a dilation that has center of
𝑐
dilation S and scale factor 𝑗
𝑐
*what happens when you multiple the original radius j by the fraction 𝑗 ? What do you get?
Circle D’ contains all the points at distance _____ from point S.
After the dilation, the image of circle D’ consists of all the points at a distance _____ from point S. But
these are exactly the points that form circle _____. Therefore the translation followed by the dilation
maps circle D onto circle S.
Since translations and dilations are ______________________________________________________
You can conclude that _________________________________________________________________
Name:__________________________
Period: ________
8.5a Similarity Lab
I. Choose a tall object with a height that would be difficult to measure directly.
1. Mark crosshairs on your mirror. The intersection will be point X. Place the mirror on the ground
several meters from your object.
2. An observer should move to point P in line with the object and the mirror in order to see the reflection
of an identifiable point F at the top of the object at point X on the mirror.
3. Measure the distance PX and PB (B directly below point F). Measure the distance from P to the
observer’s eye level E.
4. a) Why is ∠B ≈∠P ?_____________________________________________________________
b) Name the two similar triangles Δ _______ ~ Δ _________
c) By which similarity postulate are the two triangles similar? (AA~, SSS~ or SAS~) _____________
Why?
5. Set up a proportion using corresponding sides of similar triangles. Use it to calculate FB, the height of
the tall object. Show your work here:
6. Discuss possible causes of error in this experiment:
III. Another indirect method is using shadows.
Problem set.
1. A flagpole is 4 meters tall and casts a 6-meter long shadow. At the same time of day, a nearby
building casts a 24 meter shadow. How tall is the building?
2. A surveyor used the map to the right to find the distance across Lake Okeechobee.
Write and solve a proportion to find the distance across Lake Okeechobee.
Explain why ΔABC is similar to ΔADE
3.
Name: ______________________________Period:
8.5 b Similar Triangles
1. A statue, honoring Ray Hnatyshyn (1934–2002), can be found on Spadina Crescent East, near the University
Bridge in Saskatoon. Use the information below to determine the unknown height of the statue.
2. A tree 24 feet tall casts a shadow 12 feet long. Brad is 6 feet tall. How long is Brad's shadow? (draw a diagram
and solve)
3. Triangles EFG and QRS are similar. The length of the sides of EFG are 144, 128, and 112. The length of the
smallest side of QRS is 280, what is the length of the longest side of QRS? (draw a diagram and solve)
4. A 40-foot flagpole casts a 25-foot shadow. Find the shadow cast by a nearby building 200 feet tall. (draw a
diagram and solve)
5. A girl 160 cm tall, stands 360 cm from a lamp post at night. Her shadow from the light is 90 cm long. How
high is the lamp post?
hint: how long is the lamppost shadow?
160 cm
90 cm
360 cm
6. A tower casts a shadow 7 m long. A vertical stick casts a shadow 0.6 m long. If the stick is 1.2 m high, how
high is the tower? (draw a diagram and solve)
7. Triangles IJK and TUV are similar. The length of the sides of ΔIJK are 40, 50, and 24. The length of the
longest side of ΔTUV is 275, what is the perimeter of ΔTUV? (draw a diagram and solve)
8. A tree with a height of 4m casts a shadow 15 m long on the ground. How high is another tree that casts a
shadow which is 20 m long? (draw a diagram and solve)
9. Triangles CDE and NOP are similar. The perimeter of smaller triangle ΔCDE is 133. The lengths of two
corresponding sides on the triangles are 53 and 212. What is the perimeter of Δ NOP?
Name: _________________________
Period: ________
8.6 Triangle Proportionality
Show your work.
1. a = ____
2. h = ____ k = ____
3. m = ____
4. m = _____
n = _____
5. Is ΔABC ~ ΔPQR?
Explain why or why not
6. Is ΔPDQ ~ ΔLDT?
Explain why or why not
7. 𝑇𝐴 ||𝑈𝑅
𝐼𝑠 ∠𝑄𝑇𝐴 ≅ ∠𝑇𝑈𝑅? ________ Why?
8. Find x and y
𝐼𝑠 ∠𝑄𝐴𝑇 ≅ ∠𝐴𝑅𝑈? ________ Why?
Why is ΔQTA ~ ΔQUR?
e = ____
Q
4 A
3
T
e
(x, 30)
(15, y)
(5,3)
R
5
x = _______
y = _______
U
Name: ___________________________ Period: ___________
8.7 Scaled up Pythagoras
1) Begin by drawing a right triangle using the right angle in the top left corner.
2) Label the hypotenuse “c” and the other two angles A and B (∠𝐵 across from side b).
3) Measure each side of the original triangle. Use centimeters. Attend to precision. Record in table.
4) Use the length of a as your first scale factor. Record in table. Draw the original triangle scaled up by
the factor of a. You may use the exact lengths to draw the triangle but label the new triangle in terms of
a, b, c and the correct angles A, B, C.
5) Repeat step 4 with two additional triangles using b and then c as your scale factor, record in table.
6) Label all the sides and all the angles on the interior of the triangle. Use the letters not the centimeter
length. For example, this means sides are labeled ab, b2, and bc and not using numbers.
7) Cut out the three scaled up triangles.
8) Use the three manipulatives to prove the Pythagorean Theorem.
*Note that using actually physical measurement or claiming that your proof is valid because of “lining
things up” is not enough to prove for all cases. You should focus on what kind of shapes you can create
with these three cut out triangles, and use the properties of the collaborative new shape to prove the
Pythagorean Theorem.
When you are ready to write your proof, assemble the pieces into the shape you created, and glue or tape
your pieces together in the space below. Then use the lines to write your proof of the Pythagorean
Theorem.
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
Multiply your original triangle by each of the side lengths a, b, then c. Record each dilation in the table.
Original triangle
side lengths in cm
Use the 3
right angles
to construct
the 3 dilated
versions of
your original
triangle. You
can extend
the lines if
needed.
LABEL
each
hypotenuse
and AND
ANGLES
A,B,C on
the inside
of the Δs.
Cut out your
3 dilated
triangles to
use with
“Scaled Up
Pythagoras.”
a
a = ______ Scale Factor
record new side lengths
a2 =
b = ______ Scale Factor
record new side lengths
ab =
c = ______ Scale Factor
record new side lengths
ac =
b
ab =
b2 =
bc =
c*
ac =
bc =
c2=
* verify with Pythagorean Theorem
“a” scale factor
“b” scale factor
“a” scale factor
“b” scale factor
“c” scale factor
Name: ______________________
Period: __________
8.8 Exam Review
1. Use the point A as the center of dilation, then draw the dilation using the scale factor of 3
2. Dilate the polygon with vertices
A(4,1), B(2,3), C(-3,4), D (-4,-4), E(1,-3)
from the origin by a scale factor of 1.5
3. Dilate the polygon with vertices
A(8,10), B(-4,6), C(-6,7), D(5,8)
2
from the origin by a scale factor of .
5
y
y
10
9
8
7
6
5
4
3
2
1
–10–9 –8 –7 –6 –5 –4 –3 –2–1
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
x
–10–9 –8 –7 –6 –5 –4 –3 –2–1
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
1 2 3 4 5 6 7 8 9 10
x
4. Determine if the triangles are similar. If so state why, and give the scale factor from the smaller to the larger.
5. For each pair of objects determine if the two objects are similar using similarity transformations (translations, rotations,
reflections, and dilations) to establish whether the two objects are similar. On the lines, explain the transformations.
6. A hiker, whose eye level is 2 m above the ground, wants to find the height of a tree. He places a mirror horizontally on
the ground 20 m from the base of the tree, and finds that if he stands at a point C, which is 4 m from the mirror B, he can
see the reflection of the top of the tree. How tall is the tree?
7. To find the width of a river, Jordan surveys the area and finds the following measures. Find the width of the river.
8. Find the missing length.
9. Find the missing length
10. In addition to these problems you need to be able to prove the following: 1) When a segment is dilated, the image is
parallel to the pre-image. 2) All circles are similar. 3) That the Pythagorean Theorem can be obtained from similar
triangles.