4.1: Dilations and Similar Triangles Tonight’s Homework: 4.1 Handout WELCOME Chapter 4: Similarity 4.1: Dilations and Similar Triangles Tonight’s Homework: 4.1 Handout

Warm Up Solve the proportions: 1. 2.

Chapter 4 Learning Targets

Ratios a b = a : b Ratio of a to b = *Simplify when possible* Relationship between two quantities using the same units. *Simplify when possible* a b = a : b Ratio of a to b =

The ratio of the lengths for two corresponding sides Scale Factor The ratio of the lengths for two corresponding sides ABCD ∼ EFGH E 10in F A 5in B     16in 8in   D C H G

Congruence Vs. Similarity D ≅ A F E C B  

Dilation Investigation

(0,0)

Transformation Basics Figures in a plane can be reflected, rotated, or translated to produce new figures. Pre-image: The original figure Image: The new version of the figure after being transformed Transformation: The operation that maps, or moves, the pre-image onto the image

Congruence Vs. Similarity ≅ X D ∼ A M F E C B Z S P Y    

A non rigid transformation where the image and preimage are similar Dilation A non rigid transformation where the image and preimage are similar F k p F C The image is a dilation of the preimage with scale factor k:p from the center C

Reduction vs. Enlargement 0 < k < 1 Enlargement: k > 1

Requirements For Dilation Dilation with center C and scale factor K maps point P to P’, and… If P is not on C, then P’ is on CP. Also Scale factor k = (k>0 and k≠1 ) 2. If P is on C, then P=P’ CP’ CP P’ C P P = P’ C

The ratio of the lengths for two corresponding sides Scale Factor The ratio of the lengths for two corresponding sides ABCD ∼ EFGH E 10in F A 5in B     16in 8in   D C H G

“ABCD is Similar to EFGH” Similar Polygons Polygons with all corresponding angles ≌ and all sets of sides proportional B If ∠A ≌ ∠E & ∠B ≌ ∠F ∠C ≌ ∠G & ∠D ≌ ∠H Then “ABCD is Similar to EFGH” F A E H G D C ABCD ∼ EFGH

Similar Polygons Polygons with all corresponding angles ≌ and all sets of sides proportional A If ∠A ≌ ∠E & ∠B ≌ ∠F ∠C ≌ ∠G Scale Factor = Then “ABC is Similar to EFG” E G F C B ABC ∼ EFG

Dilation on Coordinate Plane When Dilating on the plane w/ Center @ (0,0) Multiply both the x and y value by the scale factor (x,y) -> (kx,ky)

Equations that equate two ratios are called proportions. b c d =

Proportion Practice

= = a x x d x a∙b Geometric Mean Given two numbers ‘a’ & ‘d’ the geometric mean is the value ‘x’ such that… and a x x d x a∙b = =