1. Dilations and Scale Factors
Section 8.1
Dilations and Scale Factors

2. 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 y-coordinates of a point by the same number, n.
The number n is called the scale factor of the transformation.

3. 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.

4. 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/ – 1
(2 · 1, 2 · 0) & (2 · 5, 2 · 3) ->
(2, 0) & (10, 6) image
m = 6 – 0 = 6/8 = 3/ – 2

5. 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 slope
x₂ - x₁
m = - 5 – = - 20/ – (- 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
y = (-5/3)x
slope of a line

6. Section 8.2
Similar Polygons

7. 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.

8. 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.

9. Polygon Similarity Show that the two polygons below are similar.
A AB = BC = AC
EF FD ED 3
E B C 3 = = 5
Each ratio is proportional.
△ABC ~ △EFD
F D ~ means similar

10. 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].

11. Section 8.3
Triangle Similarity

12. 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.

13. Triangle Similarity Prove each pair of triangles are similar.
A L D 62⁰ E M R
⁰ ⁰ ⁰
55°
J K F
< E = 180
55° < E = N
C B ° = 55° < E = 71
20 = (6) = 15(8)
= 120 proportional < D ≌ <M and < E ≌ < R
△ACB ~△LJK by SAS Similarity △DEF ~ △MRN by AA Similarity

14. Triangle Similarity Prove the two triangles are similar. X 10 T Z 10.5
Y H G
GH = TH = GT = 12
ZX YX ZY △GTH ~ △ZYX by SSS Similarity
15 = = = Three sides of one triangle are
proportional to three sides of another.

15. The Side-Splitting Theorem
Section 8.4
The Side-Splitting Theorem

16. 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.

17. Side-Splitting Theorem
Example: H
HD = DF 20 = 5
HE EG x
D E
X 20x = 22(5)
F G 20x = 110
20x = 110
x = 5.5

18. Two-Transversal Proportionality Corollary
Example:
= x x
5(3) = 9x
15 = 9x
= x

19. Indirect Measurement and Additional Similarity Theorems
Section 8.5
Indirect Measurement and Additional Similarity Theorems