NAME ______________________________________
Gr8 – MOD 3 – Lesson 8
8.G.A.4 : Understand that a 2-dimensional figure is similar to another if the second can be obtained from the first by a
sequence of rotations, reflections, and translations, and dilations; given two similar two-dimensional figures,
describe a sequence that exhibits the similarity between them.
STUDENT OUTCOMES
 Students know the definition of similarity and why dilation is not enough to determine similarity.

Given two similar figures, students describe the sequence of dilation and a congruence that would map one figure onto the other.
EXAMPLE 1
1
In the picture below, we have a triangle 𝐴𝐵𝐶 that has been dilated from center 𝑂 by a scale factor of 𝑟 = 2.
The dilated image is noted by 𝐴′𝐵′𝐶′. We also have triangle 𝐴′′𝐵′′𝐶′′, which is congruent to triangle ′𝐵′𝐶′ .
GIVEN
△ 𝐴B𝐶 ~ △ 𝐴′ 𝐵 ′ 𝐶 ′
Since we were told a dilation was performed,
we know the pre-image and image have to be
similar since dilation yields similar figures.
△ 𝐴′ 𝐵 ′ 𝐶 ′ ≅ △ 𝐴′′𝐵′′𝐶′′
These triangles were drawn congruent. This
information is given above.
FIND: Will △ 𝐴′′ 𝐵 ′′ 𝐶 ′′ ~ △ 𝐴𝐵𝐶 ?
_______
PROVE: To show △ 𝐴′′ 𝐵 ′′ 𝐶 ′′ is similar to △
𝐴𝐵𝐶 we need to find a sequence of
transformations that would map
△ 𝐴′′ 𝐵 ′′ 𝐶 ′′ to △ 𝐴𝐵𝐶. This sequence needs to be a dilation followed by a congruence (sequence of
rotation, reflection and/or translation).
FIRST ... How would you make △ 𝐴′′𝐵′′𝐶′′ congruent to △ 𝐴B𝐶 ?
STEPS:
1. Dilate
2. Translate
3. Rotate or reflect
4. Continue until the figure is mapped onto the other
5. Therefore,
Describe the sequence that would map △ 𝐴′′𝐵′′𝐶′′ onto △ 𝐴𝐵𝐶 𝑎nd prove that △ A′′ B′′ C′′ ~ △ ABC .
D ______________ △ 𝐴′′ 𝐵 ′′ 𝐶 ′′ from center O by a scale factor of r = ______. Then T_____________ the dilated
triangle _______ units ___________ and ________ units ________.
EXERCISES 1-4
1
(1) Triangle 𝐴𝐵𝐶 was dilated from center 𝑂 by scale factor 𝑟 = 2. The dilated triangle is noted by 𝐴′𝐵′𝐶′.
Another triangle 𝐴′′𝐵′′𝐶′′ is congruent to triangle 𝐴′𝐵′𝐶′ (i.e., △ 𝐴′′𝐵′′𝐶′′ ≅△ 𝐴′𝐵′𝐶′).
Describe a dilation followed by
a sequence of basic rigid
motion that would map △
𝐴′′𝐵′′𝐶′′ onto △ 𝐴𝐵𝐶.
First,
Next,
Therefore,
(2) Describe a sequence that would show △ 𝐴𝐵𝐶~ △ 𝐴′ 𝐵 ′ 𝐶 ′ .
First,
Next,
Then,
Therefore,
(3) Are the two triangles shown below similar? _________
𝑛𝑒𝑤
𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙
=
(HINT: use a proportion)
=
If yes, scale factor is: _______
If so, describe a sequence that would prove △ 𝐴𝐵𝐶~ △ 𝐴′𝐵′𝐶′.
If not, state how you know they are not similar.
EXPLANATION/PROOF:
(4) Are the two triangles shown below similar? __________
𝑛𝑒𝑤
𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙
=
=
If yes, scale factor is: _______
If so, describe a sequence that would prove △ 𝐴𝐵𝐶~ △ 𝐴′𝐵′𝐶′.
If not, state how you know they are not similar.
EXPLANATION/PROOF:
Closing
DEFINITION - Similarity is a mapping of one figure onto another figure as a sequence of a d_____________
followed by a c__________________ (a sequence of basic rigid motions).
To prove two figures are similar we need to find a s_________________ of a d_______________ followed by a
c______________________ which is a sequence of one or more b__________ r__________ m______________ .
Lesson Summary
Similarity is defined as mapping one figure onto another as a sequence of a dilation followed by a congruence (a sequence of rigid
motions).
The notation △ 𝐴𝐵𝐶~ △ 𝐴′ 𝐵′ 𝐶 ′ means that △ 𝐴𝐵𝐶 is similar to △ 𝐴′𝐵′𝐶′.