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Understanding Similarity in Geometry: Transformations and Ratios

Exploring congruence vs. similarity in figures, dilation, transformations, similarity ratios, and proving shapes similar in geometry. Learn how to identify and work with similar shapes effectively.

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Understanding Similarity in Geometry: Transformations and Ratios

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Presentation Transcript

  1. Similarity

  2. Information

  3. Congruence vs. similarity B Remember that congruent figures are identical in both shape and size. B′′ A′ A′′ B′ A C C′ C′′ △ABC ≅△A′B′C′≅△A′′B′′C′′ Two figures are called similar if they have the same shape, but not necessarily the same size. E E′′ D′ D′′ E′ Similar figures are indicated using ~ . D F′ F F′′ △DEF~△D′E′F′~△D′′E′′F′′

  4. Dilation A dilation changes the size of a figure, but not its shape. What observations can you make about the relationship between figures ABCDE and A′B′C′D′E′? 3.0 C′ B′ 2.0 B C 2.25 1.5 1.0 1.5 D D′ 0.5 0.75 E A 1.0 A′ E′ 1.5 • The lengths of the sides of A′B′C′D′E′ are 1.5 times the length of the corresponding sides of ABCDE. 1.5 is called the scale factor. • Corresponding angles are the same in both figures.

  5. Similarity transformations Does each of these transformations preserve similarity?   rotation translation   reflection dilation   stretch shear These are called similarity transformations.

  6. Similarity ratio The similarity ratio between two similar figures is the ratio of any pair of corresponding lengths on the figures. length on dilated image similarity ratio = corresponding length on pre-image What is the similarity ratio for this dilation? A′B′ B′C′ A′C′ 1.1 sim. ratio = = = A B AB BC AC 0.5 1.2 substitute values: C 2.2 2.2 2.4 1.0 A′ B′ sim. ratio = = = 1.1 1.2 0.5 2 1.0 = = 2 1 2.4 C′

  7. Scale factors between 0 and 1 What happens when the scale factor for a dilation is between 0 and 1? When the scale factor is between 0 and 1, the dilation will be smaller than the original object. These figures have similarity ratio ½. What is B′C′? B′C′ 2.2 similarity ratio = A B BC substitute values: 1.0 2.4 B′C′ B′C′ ½ = = BC 2.4 C A′ B′ solve for B′C′: B′C′ = ½ × 2.4 = 1.2 C′

  8. Find the similarity ratio

  9. Proving shapes similar To prove shapes similar, it must be shown that the figures are related by similarity transformations. What are two ways you can prove two polygons similar? by similarity transformations: P′Q′R′is PQR rotated by 180° and dilated by 13/10, so they are similar. 2.0 by proving corresponding angles congruent and corresponding lengths to be the same ratio: R Q 57° P′ 2.5 38° P′Q′ Q′R′ R′P′ 3.2 38° = = = 13/10 PQ QR RP 4.16 3.25 P mP=mP′ ⇒ P≅P′ mQ=mQ′ Q ≅ Q′ ⇒ 57° mR=mR′ R ≅ R′ R′ ⇒ Q′ 2.6

  10. Similar shapes

  11. Finding lengths in two similar shapes

  12. Summary activity

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