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13-4 Congruent and Similar Solids

13-4 Congruent and Similar Solids. Objectives:. Identify congruent or similar solids. State the properties of similar solids. . Similar Solids. Solids that have exactly the same shape but not necessarily the same size.

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13-4 Congruent and Similar Solids

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  1. 13-4 Congruent and Similar Solids

  2. Objectives: • Identify congruent or similar solids. • State the properties of similar solids.

  3. Similar Solids • Solids that have exactly the same shape but not necessarily the same size. • You can determine if two solids are similar by comparing the ratios of the corresponding linear measurements.

  4. Similar Solids 2 5 2 5 2 5 8 20 2 5 6 15 6cm 20cm 8cm The ratio of the measures is called the scale factor. 5cm 15cm 2cm

  5. Congruent Solids • Solids that are exactly the same shape and exactly the same size. • They have a scale factor of 1.

  6. Congruent Solids • Two solids are congruent if: • The corresponding angles are congruent, • The corresponding edges are congruent, • The corresponding faces are congruent, and • The volumes are equal.

  7. Example 1: 17 in 13 in 12 in 15 in 16 cm ¬ ¬ 8√ 7 cm 5 in 8 in 4√ 7 cm 8 cm ¬ ¬ 4√ 3 cm 8√ 3 cm

  8. Example 1: Cont. Determine whether the REGULAR pentagonal pyramids are similar, congruent, or neither. 16 cm 8√ 7 cm 4√ 7 cm 8 cm ¬ ¬ 4√ 3 cm 8√ 3 cm

  9. Example 1: Cont. The ratios of the measures are equal, so we can conclude that the pyramids are similar. Base edge of larger pyramid Base edge of smaller pyramid 8√ 3 cm 4√ 3 cm Height of larger pyramid Height of smaller pyramid 16 cm 8 cm 16 cm 8√ 7 cm Lateral edge of larger pyramid Lateral edge of smaller pyramid 8√ 7 cm 4√ 7 cm 4√ 7 cm 8 cm ¬ ¬ 2 4√ 3 cm 8√ 3 cm

  10. Example 1: Cont. 8 5 Radius of larger cone Radius of smaller cone Since the ratios are not the same, there is no need to find the ratio of the slant heights. The cones are not similar. 15 12 17 in 13 in 12 in 15 in Height of larger cone Height of smaller cone ¬ ¬ 5 in 8 in Determine whether the cones are similar, congruent, or neither.

  11. Theorem 13.1 If two solids are similar with a scale factor of a:b, then the surface areas have a ratio of a²:b², and the volumes have a ratio of a³:b³. • Scale Factor 3:2 • Ratio of surface areas 3²:2² or 9:4 • Ratio of volumes 3³:2³ or 27:8 9 ¬ ¬ 6 6 6 4 4

  12. Example 2: Volleyballs are spheres. One ball has a diameter of 4 inches, and another has a diameter of 20 inches. Find the scale factor of the two volleyballs. 1 5 Diameter of the smaller sphere Diameter of the larger sphere 4 20 The scale factor is 1:5 Simplify

  13. Example 2: Cont. The ratio of the surface areas is 1:25 Find the ratio of the surface areas of the two spheres. a² 1² b² 5² Surface area of smaller sphere Surface area of larger sphere 1 Simplify 25

  14. Example 2: Cont. The ratio of the volumes of the two spheres is 1:125. Find the ratio of the volumes of the two spheres. Volume of the smaller sphere Volume of the larger sphere 1³ 5³ a³ b³ 1 Simplify 125

  15. Assignment: • Pg. 710 • 5-16,18-23, 27-30 ALL By: Kristen Miller and Erin Fields 1st hr.

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