Teacher’s Notes
Triangle Similarity: Dilation Guided Notes
Dilation means to expand or shrink a figure. It is a type of transformation that changes the size of the
image.
Scale factor tells you by how much you need to expand or shrink the figure.
When the scale factor is greater than 1, then the figure gets bigger.
When the scale factor is less than 1, then the figure gets smaller.
Examples:
Write in the blank whether the figure would be expanding or shrinking according to the scale factor.
1. 3
expanding
2. ½
shrinking
3. 6.5
expanding
4. ¾
shrinking
5. 4
expanding
6. 2
expanding
7. 10,000
expanding
8. 0.1
shrinking
Prime notation is used to distinguish the image from the pre-image.
Prime notation is marked by using this symbol: ‘.
Example:
1.
Teacher’s Notes
In order to dilate points on a coordinate plane, you must multiply the x and y by the scale factor.
In this example, you need to perform a Dilation of 4 (D4). You need to multiply each coordinate of the
original point (called the image) by the scale factor which is 4.
1.
2.
3.
4.
5.
Determine scale factor: 4
Determine point: A (2, 3)
Multiply each coordinate by the scale factor: A (2*4, 3*4)
Simplify: A’ (8, 12)
Plot
Multiply both coordinates by
scale factor
(3*1/3, 6*1/3)
Simplify
(1, 2)
Graph
Teacher’s Notes
Multiply
both
coordinates
by scale
factor
Simplify
A (2*1/2, 2*1/2)
B (4*1/2, 6*1/2)
C (-2*1/2, 4*1/2)
A (1, 1)
B (2, 3)
C (-1, 2)
Graph
Practice (remember to use prime notation and SHOW WORK):
1. X (1, 3) Y (2, 4) Z (3, 5) with Dilation of 5.
X’ (5, 15) Y’ (10, 20) Z’ (15, 25)
2. E’ (2, 3) F’ (4, 6) G’ (-4, -8), scale factor = ½.
E (1, 1.5) F (2, 3) G (-2, -4)
1
3
3. F’ (-3, 9) E’ (6, -12) T’ (-6, -3), scale factor = .
F (-1, 3) E (2, -4) T (-2, -1)
4. S (5, 7) T (3.5, 5) U (-2, -4.5), scale factor = 2.
S’ (10, 14) T’ (7, 10) U’ (-4, -9)
Teacher’s Notes
Vocabulary
1. Base angles – angles formed by the base and legs of a triangle; they are always equal.
2. Corresponding parts – angles/sides in the same location.
𝐼𝑛 ∆𝐴𝐵𝐶 𝑎𝑛𝑑 ∆𝑋𝑌𝑍,
∠𝐴 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑠 𝑤𝑖𝑡ℎ ∠𝑋
𝐼𝑛 𝑞𝑢𝑎𝑑𝑟𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙𝑠 𝐽𝐾𝐿𝑀 𝑎𝑛𝑑 𝑅𝑆𝑇𝑈,
𝐽𝐾 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑠 𝑤𝑖𝑡ℎ 𝑅𝑆.
3. Dilation – a type of transformation that changes the size of a shape by expanding or shrinking.
4. Legs – In a right triangle, they are the shorter sides which meet to form the right angle. In an
isosceles triangle, they are the two congruent sides.
5. Proportionality – something changed by the same ratio
𝑠𝑖𝑑𝑒 𝐴
𝑠𝑖𝑑𝑒 𝐵
𝑠𝑖𝑑𝑒 𝑋
= 𝑠𝑖𝑑𝑒 𝑌
6. Scale factor – ratio of change; the amount by which a shape needs to expand/shrink; dilation
7. Similarity – the same 'shape' but are just scaled differently. Similar triangles have congruent
angles and proportional sides.
**Notation: △ABC ~△XYZ means that "△ABC is similar to △XYZ "
Teacher’s Notes
Practice
1. If ∆ABC ~ ∆UYT, then what sides/angles correspond with:
a. AB: ∆ABC ~∆UYT, so AB corresponds with UY
b. ∠𝐵𝐶𝐴: ∆ABC ~∆UYT, so ∠𝐵𝐶𝐴 corresponds with ∠𝑌𝑇𝑈
c. TU: ∆ABC ~∆UYT, so TU corresponds with CA
d. ∠𝑇𝑈𝑌: ∆ABC ~∆UYT, so ∠𝑇𝑈𝑌 corresponds with ∠𝐶𝐴𝐵
**Sides of similar triangles are proportional.
**Angles of similar triangles are congruent.
Finding the scale factor/similarity ratio/dilation
What is the scale factor for∆𝐴𝐵𝐶 𝑎𝑛𝑑 ∆𝑋𝑌𝑍? ______
If △ABC ~△WXY , then what is the
similarity ratio?
http://www.mathwarehouse.com/geo
metry/similar/triangles/sides-andangles-of-similar-triangles.php
http://www.ck12.org/geometry/Dilati
on-in-the-CoordinatePlane/lesson/Dilation-in-theCoordinate-Plane-Intermediate/
1. Pick a pair of corresponding
sides (follow the letters )
AB and WX are corresponding.
Follow the letters: △ABC ~
△WXY
2. Substitute side lengths into
proportion
AB/WX = 7/21
3. Simplify (if necessary)
7/21 = 1/3
Teacher’s Notes
1. Why is the following problem unsolvable?
If △ JKL ~ △ XYZ, LJ = 22, JK = 20 and YZ = 30, what is the similarity ratio?
Answer: You are not given a single pair of corresponding sides so you cannot find the similarity ratio.
Corresponding sides follow the same letter order as the triangle name so
YZ of △XYZ corresponds with side KL of△JKL
JK of △JKL corresponds with side XY of△XYZ
LJ of △JKL corresponds with side ZX of△XYZ
Below is a picture of what these two triangles could look like:
2. If △ ABC ~ △ ADE, AB = 20 and AD = 30, what is the similarity ratio?
Step 1) Pick a pair of corresponding sides
AB and AD
(follow the letters )
△ABC ~ △ADE
Step 2) Substitute side lengths into
𝑨𝑩 𝟐𝟎
=
𝑨𝑫 𝟑𝟎
proportion
Step 3) Simplify (if necessary)
𝟐𝟎 𝟐
=
𝟑𝟎 𝟑
a. If EA = 33, how long is CA?
i. EA and CA are corresponding sides ( △ABC ~ △ADE )
Since the sides of similar triangles are proportional, just set up a proportion
involving these two sides and the similarity ratio and solve.
𝑬𝑨 𝟑
𝟑𝟑 𝟑
𝟔𝟔
= →
=
→ 𝑪𝑨(𝟑) = 𝟑𝟑(𝟐) → 𝑪𝑨(𝟑) = 𝟔𝟔 → 𝑪𝑨 =
→ 𝑪 = 𝟐𝟐
𝑪𝑨 𝟐
𝑪𝑨 𝟐
𝟑
Similarity


There are 3 types of similarity: AA, SAS, and SSS
Triangles are similar if:
o All of their angles are equal
o Their corresponding sides are in the same ratio
A triangle ALWAYS
equals 180°
Teacher’s Notes
I.
AA Similarity (angle-angle)
a. Triangles have two matching angle values
b. If the triangles have two angles in common, then they always have all three in common
i.
II.
SSS Similarity (side-side)
a. Triangles have 3 pairs of sides in the same ratio
i.
III.
SAS Similarity (
)
a. Triangles have two pairs of sides in the same ratio and the included angles are also
equal
i.