Download ​ Similar​figures are figures that are the same shape, but not

Unit 5: Similarity
Lesson 3: Similar Figures
In our study of transformations, we have seen many figures that remain congruent after translations,
reflections, and rotations [these were called rigid transformations]. We have also seen figures which
retain their shape but do not remain the same size after applying a size transformation (also called a
dilation). These figures and their images after a size transformation are similar figures.
Definition: ​ Two figures are ​similar​ if one is the image of the other under a
size transformation of scale ​ factor, ​k, or under a composite of
transformations (one of which much be a size change). That is to
say, one figure is a ​dilation​ of the other.
Similar​ figures are figures that are the same shape, but
not necessarily the same size.
If these figures should also be the same size, the figures
are called ​congruent​.
Similar triangles - scale factor 2
Are the figures below similar figures?
1. Are the figures the same shape?
____________
2. Do they appear to have the same angles?
____________
3. What do you observe is different between the two triangles?
In order to be similar, we need to be able to perform transformations (translations, rotations,
reflections, and dilations) on one triangle to create the other.
4. What are the coordinates of ΔABC?
A _______
B_______
C_______
5. Reflect ΔABC over the y-axis. What are
the coordinates of the reflected points?
A' _______ B'_______
C'_______
6. Now plot the points for the reflected triangle A'B'C'.
7. Now focus on ΔA'B'C' and ΔLMN. Name pairs of corresponding angles.
______ & ______
______ & ______
______ & ______
8. Comparing the coordinates of the corresponding angles, what appears to be the scale
factor applied to ΔA'B'C'?
We now have
​
similar triangles since we could apply transformations that ​mapped ΔABC
onto ΔNML
9. Fill in the blanks to summarize the above:
Starting with ΔABC, we first _________ over the ________. Then we
_____________ by a scale factor of _______ . Now, writing this as a
similarity transformation rule: (x, y) ​→​ ( ______ , _____ )
10. You are told to perform a similarity transformation that is the composition of the following two
transformations (performed in the order given)
a. Describe the following transformations:
Transformation I:
(x, y)→(23x, 23y)
Transformation II: (x, y)→(x,− y)
A size change of scale factor ________
A reflection over the _______________.
b. ΔPQR has vertices P(6, 12), Q(18, 0), R(12, ​−​6). Using the transformations from part (a),
what would be the vertices of the final image?
P (6, 12)
Q (18, 0)
R (12, -6)
P' ________
Q' ________
R' ________
P'' ________
Q'' ________
R'' ________
c. Write a rule that combines the two transformations from above into a similarity
transformation.
(x, y) ​→​ ( ______ , ______ )
Two polygons with the same number of sides are ​SIMILAR ​provided that
1) Their corresponding angles have the same measure and
2) The ratios of lengths of corresponding sides is a constant (this constant is the s​ cale
factor).
In the above diagram, quadrilateral ​A'B'C'D' ~ quadrilateral A
​ BCD.
The symbol ~ means “is similar to.”
Proof that they are similar:
1) Corresponding angles have the same measure:
m​∠​ ​A'= m​∠​ ​A
m​∠​ ​B'= m​∠​ ​B
m​∠​ ​C'= m​∠​ ​C
m​∠​ ​D'= m​∠​ ​D
2) Ratios of lengths of corresponding sides is a constant:
The constant
5
2
is called the ​scale factor​ from quadrilateral ​ABCD to quadrilateral A
​ 'B'C'D'. It scales
(multiplies) the length of each side of quadrilateral ​ABCD to produce the length of the corresponding
side of quadrilateral ​A'B'C'D'.
Questions:
1. If the scale factor from A
​ BCD to ​A'B'C'D' is 5/2, what is the scale factor from ​A'B'C'D' to
ABCD?
2. If two figures are similar, how can we find the scale factor from the smaller figure to the larger
figure?
3. Follow up question: how do we find the scale factor from the larger figure to the smaller figure?
4. Suppose Δ​PQR ~ ΔXYZ and the scale factor from Δ​PQR to ​ΔXYZ is 34 . What is true about the
corresponding angles? What is true about pairs of corresponding sides? Be specific and list
corresponding angles and sides in your answer below.
State if the polygons are similar by finding the ratios of corresponding sides.
The polygons in each pair are similar. Find the missing side length.
The polygons in each pair are similar. Solve for x.
The polygons in each pair are similar. Find the missing side length.
The polygons in each pair are similar. Solve for x.
Knowing that two triangles are similar allows you to conclude that the three pairs of corresponding
angles are congruent, and that the three pairs of corresponding sides are related by the same scale
factor. Conversely, if you know that the three pairs of corresponding angles are congruent and the
three pairs of corresponding sides are related by some scale factor, you can conclude that the triangles
are similar!
33. Each triangle described in the table below is similar to ΔABC. For each triangle (ΔDEF, ΔGHI,
ΔJKL), use this fact and the additional information given to fill in the table.
Triangle Angle Measures
ΔABC
m​∠​ A =
64°
m​∠​ B =
18°
m​∠​ C =
98°
ΔDEF
m​∠​ D =
____°
m​∠​ E = 64°
m​∠​ F = 18°
Triangle Angle Measures
ΔAB
C
ΔGH
I
m​∠​ A =
64°
m​∠​ B = 18°
m​∠​ C =
98°
m​∠​ G =
____°
m​∠​ H =
____°
m​∠​I =
____°
Triangle Angle Measures
ΔAB
C
ΔJK
L
Shortest
Side
Length
Longest
Side
Length
Third
Side
Length
AC = 4.0
AB = 12.8
BC = 11.6
Scale
Factor
from
ΔABC
2
Shortest
Side
Length
AC = 4.0
Longest
Side
Length
AB = 12.8
IG = 6.4
Shortest
Side
Length
m​∠​ A =
64°
m​∠​ B = 18°
m​∠​ C =
98°
AC = 4.0
m​∠​ J =
____°
m​∠​ K = 18°
m​∠​ L = 98°
JL = 14.0
Longest
Side
Length
AB = 12.8
Third
Side
Length
Scale
Factor
from
ΔABC
BC =
11.6
GH =
5.8
Third
Side
Length
BC =
11.6
Scale
Factor
from
ΔABC