Section 8.1

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# Section 8.1 - PowerPoint PPT Presentation

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 .

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### 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 y-coordinates 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 slope

x₂ - x₁

• 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

5 EF FD ED

3

E B 4 C 3 = 4 = 5

9 12 15

915

Each ratio is proportional.

△ABC ~ △EFD

F12 D ~ 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).
• 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 D 62⁰ E M R

8 47⁰62⁰ 71⁰

55°

20 J 6 K F

62 + 47 + < E = 180

55°109 + < E = 180 N

C 15 B 55° = 55° < E = 71

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 10.5

7 8 12

Y H

G 15

GH = 15TH = 10.5GT = 12

ZX 10 YX 7 ZY 8 △GTH ~ △ZYX by SSS Similarity

15 = 1.510.5 = 1.512 = 1.5 Three sides of one triangle are

10 7 8 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 22 HD = DF20 = 5

HE EG 22 x

D E

5 X 20x = 22(5)

F G 20x = 110

20x = 110

20 20

x = 5.5

Two-Transversal Proportionality Corollary
• Example:

5 9 5 = x

9 3

x 3

5(3) = 9x

15 = 9x

15 = 9x

9 9

1.66 = x

### Section 8.5

Indirect Measurement and Additional Similarity Theorems