Section 8.1
Dilations and Scale Factors
Dilations
• A dilation is an example of a transformation
that is not rigid. Dilations preserve the shape
of an object, but they change the size.
• A dilation of a point in a coordinate plane can
be found by multiplying the x- and ycoordinates of a point by the same number, n.
• The number n is called the scale factor of the
transformation.
Dilations
• What are the images of the points (2, 3) and
(-4, -1) transformed by the dilation of
D(x,y) = (3x, 3y)?
• (3 · 2, 3 · 3)
[3 · (- 4), 3 · (- 1)]
• (6, 9) image
(- 12, - 3) image
• The scale factor is the multiplier 3.
Using Dilations
• The endpoints of a
segment (1, 0) and
(5, 3) and a scale factor
of 2 is given.
• Show that the dilation
image of the segment
has the same slope as
the pre-image.
• m = y₂ - y₁
slope
x₂ - x₁
• m = 3 – 0 = 3/4
5–1
(2 · 1, 2 · 0) & (2 · 5, 2 · 3) ->
(2, 0) & (10, 6) image
• m = 6 – 0 = 6/8 = 3/4
10 – 2
Using Dilations
• Find the line that passes
through the pre-image
point (3, - 5) and the
image that is found by a
scale factor of – 3.
[- 3 · 3, - 3 · (- 5)] ->
(- 9, 15) image
m = y₂ - y₁
x₂ - x₁
slope
• m = - 5 – 15 = - 20/ 12
3 – (- 9)
• m = - 5/3
• y – y₁ = m(x – x₁)
point-slope form
•
•
•
•
•
•
y – 15 = (-5/3)(x – (- 9))
y – 15 = (-5/3)(x + 9)
y – 15 = (-5/3)x – 15
+ 15
+ 15
y = (-5/3)x
slope of a line
Section 8.2
Similar Polygons
Similar Polygons
• Two figures are similar if and only if one is
congruent to the image of the other by a
dilation.
• Similar figures have the same shape but not
necessarily the same size.
Polygon Similarity Postulate
• Two polygons are similar if and only if there is
a way of setting up a correspondence
between their sides and angles such that the
following conditions are met:
• Each pair of corresponding angles are
congruent.
• Each pair of corresponding sides are
proportional.
Polygon Similarity
• Show that the two polygons below are similar.
A
AB = BC = AC
EF
FD
ED
5
3
E
B
4
9
F
C
3 = 4 = 5
9
12 15
15
12
D
Each ratio is proportional.
△ABC ~ △EFD
~ means similar
Properties of Proportions
•
•
•
•
•
•
•
•
•
Let a, b, c, and d be any real numbers.
Cross-Multiplication Property
If (a/b) = (c/d) and b and d ≠ 0, then ad = bc
Reciprocal Property
If (a/b) = (c/d) and a, b, c, and d ≠ 0, then (b/a) = (d/c).
Exchange Property
If (a/b) = (c/d) and a, b, c, and d ≠ 0, then (a/c) = (b/d).
“Add-One” Property
If (a/b) = (c/d) and b and d ≠ 0, then
[(a + b)/b] = [(c + d)/d].
Section 8.3
Triangle Similarity
Triangle Similarity
• AA (Angle-Angle) Similarity Postulate:
• If two angles of one triangle are congruent to two
angles of another triangle, then the triangles are
similar.
• SSS (Side-Side-Side) Similarity Theorem:
• If the three sides of one triangle are proportional to
the three sides of another triangle, then the triangles
are similar.
• SAS (Side-Angle-Side) Similarity Theorem:
• If two sides of one triangle are proportional to two
sides of another triangle and their included angles are
congruent, then the triangles are similar.
Triangle Similarity
• Prove each pair of triangles are similar.
A
L
8
20
55°
J 6
K
D 62⁰
47⁰
E
M
F
62 + 47 + < E = 180
109 + < E = 180
< E = 71
R
62⁰
71⁰
55°
N
C
15 B
55° = 55°
20 = 8 20(6) = 15(8)
15 6
120 = 120 proportional < D ≌ <M and < E ≌ < R
△ACB ~△LJK by SAS Similarity
△DEF ~ △MRN by AA Similarity
Triangle Similarity
• Prove the two triangles are similar.
X
10
T
Z
7
8
10.5
12
Y
G
GH = 15 TH = 10.5 GT = 12
ZX 10
YX
7
ZY
8
15 = 1.5 10.5 = 1.5
12 = 1.5
10
7
8
H
15
△GTH ~ △ZYX by SSS Similarity
Three sides of one triangle are
proportional to three sides of
another.
Section 8.4
The Side-Splitting Theorem
Side-Splitting Theorem
• A line parallel to one side of the triangle
divides the other two sides proportionally.
• Two-Transversal Proportionality Corollary
• Three or more parallel lines divide two
intersecting transversals proportionally.
Side-Splitting Theorem
• Example:
H
20
D
5
F
22
HD = DF
HE
EG
20 = 5
22
x
E
X
G
20x
20x
20x
20
x
= 22(5)
= 110
= 110
20
= 5.5
Two-Transversal Proportionality
Corollary
• Example:
5
x
9
5 = x
9
3
3
5(3) =
15 =
15 =
9
1.66 =
9x
9x
9x
9
x
Section 8.5
Indirect Measurement and Additional
Similarity Theorems