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Similarity

Similarity. B. A. Similar shapes are always in proportion to each other. There is no distortion between them. C. D. A. B. C. D. Conditions for similarity. Two shapes are similar only when: Corresponding sides are in proportion and Corresponding angles are equal.

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Similarity

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  1. Similarity B A Similar shapes are always in proportion to each other. There is no distortion between them. C D A B C D

  2. Conditions for similarity • Two shapes are similar only when: • Corresponding sides are in proportion and • Corresponding angles are equal All regular polygons are similar

  3. Conditions for similarity • Two shapes are similar only when: • Corresponding sides are in proportion and • Corresponding angles are equal All rectangles are not similar to one another since only condition 2 is true.

  4. If two objects are similar then one is an enlargement of the other The rectangles below are similar: Find the scale factor of enlargement that maps A to B 8 cm Not to scale! A 5 cm 16 cm 10 cm B Scale factor = x2 Note that B to A would be x ½

  5. If two objects are similar then one is an enlargement of the other The rectangles below are similar: Find the scale factor of enlargement that maps A to B 8 cm Not to scale! A 5 cm 12 cm 7½ cm B Scale factor = x1½ Note that B to A would be x 2/3

  6. If we are told that two objects are similar and we can find the scale factor of enlargement then we can calculate the value of an unknown side. 13½ cm The 3 rectangles are similar. Find the unknown sides x cm Not to scale! 3 cm y cm C B 24 cm A 8 cm Comparing corresponding sides in A and B: 24/8 = 3 so x = 3 x 3 = 9 cm Comparing corresponding sides in A and C: 13½ /3 = 4½ so y = 4 ½ x 8 = 36 cm

  7. If we are told that two objects are similar and we can find the scale factor of enlargement then we can calculate the value of an unknown side. y cm The 3 rectangles are similar. Find the unknown sides 7.14 cm Not to scale! 2.1 cm 26.88 cm C B x cm A 5.6 cm Comparing corresponding sides in A and B: 7.14/2.1 = 3.4 so x = 3.4 x 5.6 = 19.04 cm Comparing corresponding sides in A and C: 26.88/5.6 = 4.8 so y = 4.8 x 2.1 = 10.08 cm

  8. Similar Triangles Similar triangles are important in mathematics and their application can be used to solve a wide variety of problems. • The two conditions for similarity between shapes as we have seen earlier are: • Corresponding sides are in proportion and • Corresponding angles are equal Triangles are the exception to this rule. only the second condition is needed • Two triangles are similar if their • Corresponding angles are equal

  9. These two triangles are similar since they are equiangular. 65o 70o 45o 70o 45o 50o 50o 55o 75o These two triangles are similar since they are equiangular. If 2 triangles have 2 angles the same then they must be equiangular = 180 – 125 = 55

  10. Finding Unknown sides (1) 20 cm b c 6 cm 12 cm 18 cm Since the triangles are equiangular they are similar. So comparing corresponding sides.

  11. 24 cm y x 6 cm 8 cm 10 cm Comparing corresponding sides

  12. Determining similarity A B Triangles ABC and DEC are similar. Why? C Angle ACB = angle ECD (Vertically Opposite) Angle ABC = angle DEC (Alt angles) D Angle BAC = angle EDC (Alt angles) E we know that corresponding sides are in proportion Since ABC is similar to DEC ACDC BCEC ABDE The order of the lettering is important in order to show which pairs of sides correspond.

  13. A If BC is parallel to DE, explain why triangles ABC and ADE are similar B C Angle BAC = angle DAE (common to both triangles) E D Angle ABC = angle ADE (corresponding angles between parallels) Angle ACB = angle AED (corresponding angles between parallels) A A B B E D E D C C A line drawn parallel to any side of a triangle produces 2 similar triangles. Triangles DBC and DAE are similar Triangles EBC and EAD are similar

  14. In the diagram below BE is parallel to CD and all measurements are as shown. • Calculate the length CD • Calculate the perimeter of the Trapezium EBCD A A 6 m 4.5 m 10 m E B 4.8 m 4 m D C 8 cm C 8 cm D 7.5 cm 3 cm So perimeter = 3 + 8 + 4 + 4.8 = 19.8 cm

  15. In the diagram below BE is parallel to CD and all measurements are as shown. • Calculate the length CD • Calculate the perimeter of the Trapezium EBCD Alternative approach to the same problem A A 6 m 4.5 m 10 m E B 4.8 m 4 m D C 8 cm C 8 cm D 7.5 cm 3 cm SF = 10/6 = 1.666.. So CD = 1.666… x 4.8 = 8 cm So BC = 3 cm AC = 1.666… x 4.5 = 7.5 Perimeter = 3 + 8 + 4 + 4.8 = 19.8 cm

  16. The two triangles below are similar: Find the distance y. C B 20 cm y A E D 5 cm 45 cm

  17. The two triangles below are similar: Find the distance y. C Alternate approach to the same problem B 20 cm y A E D 5 cm 45 cm So y = 20/10 = 2 cm Scale Factor = 50/5 = 10

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