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 = y - y ₂ ₁ slope x - x ₂ ₁• 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 = y - y ₂ ₁ slope x - x ₂ ₁

• m = - 5 – 15 = - 20/ 12 3 – (- 9) • m = - 5/3 • y – y = m(x – x ) ₁ ₁point-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 9 15

Each ratio is proportional.△ABC ~ △EFD F 12 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).• “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 D 62⁰ E M R 8 47⁰ 62 71⁰ ⁰ 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 = 15 TH = 10.5 GT = 12ZX 10 YX 7 ZY 8 GTH ~ ZYX by SSS Similarity△ △ 15 = 1.5 10.5 = 1.5 12 = 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 = DF 20 = 5

HE EG 22 x D E5 X 20x = 22(5)F G 20x = 110 20x = 110

20 20 x = 5.5

Two-Transversal Proportionality Corollary

• Example:

5 9 5 = x9 3

x 35(3) = 9x 15 = 9x

15 = 9x 9 9

1.66 = x

Section 8.5

Indirect Measurement and Additional Similarity Theorems