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Understanding Similarity in Shapes: Ratios, Angles, and Enlargement

Learn about similar polygons, dilation, and methods to prove similarity in shapes. Solve problems involving ratios, corresponding sides, and angles to deepen your understanding of geometric similarity.

davidmthomas
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Understanding Similarity in Shapes: Ratios, Angles, and Enlargement

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  1. WARM UP Solve for x X + 1 = X X 1 You will have to use the Quadratic Formula!

  2. 8.2 Similarity Definition: Similar polygons are polygons in which: • The ratios of the measures of corresponding sides are equal. • Corresponding angles are congruent.

  3. Similar figures: figures that have the same shape but not necessarily the same size. Dilation: when a figure is enlarged to be similar to another figure. Reduction: when a figure is made smaller it also produces similar figures.

  4. Proving shapes similar: • Similar shapes will have the ratio of all corresponding sides equal. • Similar shapes will have all pairs of corresponding angles congruent.

  5. Example: ∆ABC ~ ∆DEF D A 8 12 6 4 C E F B 5 10 Therefore: A corresponds to D, B corresponds to E, and C corresponds to F. • The ratios of the measures of all pairs of corresponding sides are equal. = = =

  6. Each pair of corresponding angles are congruent. <B <E <A <D <C <F

  7. ∆MCN is a dilation of ∆MED, with an enlargement ratio of 2:1 for each pair of corresponding sides. Find the lengths of the sides of ∆MCN. (0,8) C (0,4) E N M D (0,0) (3,0) (6,0)

  8. Given: ABCD ~ EFGH, with measures shown. 1. Find FG, GH, and EH. B 6 F 9 4 A A E C 7 3 D G H 2. Find the ratio of the perimeter of ABCD to the perimeter of EFGH.

  9. T61: The ratio of the perimeters of two similar polygons equals the ratio of any pair of corresponding sides. Given that ∆JHK ~ ∆POM, <H = 90, <J = 40, m<M = x+5, and m<O = y, find the values of x and y. First draw and identify corresponding angles.

  10. K M J P O H <J comp. <K  <K = 50 <K = <M 50 = x + 5 45 = x

  11. <H = <O 90 = y 180 = y

  12. Given ∆BAT ~ ∆DOT, OT = 15, BT = 12, TD = 9 Find the value of x(AO). A x O 15 D B 9 12 T Hint: set up and use Means-Extremes Product Theorem.

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