1 / 28

Similarity and Dilations in Triangles

Learn about dilations and similarity in triangles, including ratios, scale factors, congruence vs. similarity, and geometric mean. Explore transformations, reduction vs. enlargement, and requirements for dilation.

ahubbard
Download Presentation

Similarity and Dilations in Triangles

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. Warm Up Solve the proportions: 1. 2.

  3. Chapter 4 Learning Targets

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

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

  6. Congruence Vs. Similarity D ≅ A F E C B

  7. Dilation Investigation

  8. (0,0)

  9. 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

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

  11. 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 centerC

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

  13. 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

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

  15. 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

  16. 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

  17. 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)

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

  19. Proportion Practice

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

More Related