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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 .

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section 8 1

Section 8.1

Dilations and Scale Factors

dilations
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.
dilations1
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
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 dilations1
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

Section 8.2

Similar Polygons

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
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
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
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

Section 8.3

Triangle Similarity

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 similarity1
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 similarity2
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

Section 8.4

The Side-Splitting Theorem

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 theorem1
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
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

Section 8.5

Indirect Measurement and Additional Similarity Theorems