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similarity.geometric.shapes.mathematics.ppt
- 1. Similar Shapes Outcomes To understand the principles of similar shapes (especially triangles) and apply these to a range of questions at GCSE level
- 2. A B C D A B C D Similarity Remember, similar shapes are always in proportion to each other. There is no distortion between them.
- 4. Conditions for similarity Two shapes are similar only when: •Corresponding sides are in proportion and •Corresponding angles are equal All regular polygons are similar
- 5. 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.
- 6. 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 A B 8 cm 16 cm 5 cm 10 cm Not to scale! Scale factor = x2. (Note that B to A would be x ½)
- 7. 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 A B 8 cm 12 cm 5 cm 7½ cm Not to scale! Scale factor = x1½ (Note that B to A would be x 2/3)
- 8. If we are told that two objects are similar and we can find the scale factor of enlargement by comparing corresponding sides then we can calculate the value of an unknown side. 8 cm A B C 2 cm Not to scale! 24 cm p cm q cm 12½ cm The 3 rectangles are similar. Find the unknown sides, p and q 1. Comparing corresponding sides in A and B. SF = 24/8 = x3. 2. Apply the scale factor to find the unknown side. p = 3 x 2 = 6 cm.
- 9. If we are told that two objects are similar and we can find the scale factor of enlargement by comparing corresponding sides then we can calculate the value of an unknown side. 8 cm A B C 2 cm Not to scale! 24 cm p cm q cm 12½ cm The 3 rectangles are similar. Find the unknown sides, p and q. 1. Comparing corresponding sides in A and C. SF = 12.5/2 = x6.25. 2. Apply the scale factor to find the unknown side. q = 6.25 x 8 = 50 cm.
- 10. 5 cm If we are told that two objects are similar and we can find the scale factor of enlargement by comparing corresponding sides then we can calculate the value of an unknown side. A B C 2.1 cm Not to scale! p cm 7.14 cm 35.5 cm q cm The 3 rectangles are similar. Find the unknown sides p and q 1. Comparing corresponding sides in A and B. SF = 7.14/2.1 = x3.4. 2. Apply the scale factor to find the unknown side. p = 3.4 x 5 = 17 cm.
- 11. 5 cm If we are told that two objects are similar and we can find the scale factor of enlargement by comparing corresponding sides then we can calculate the value of an unknown side. A B C 2.1 cm Not to scale! p cm 7.14 cm 35.5 cm q cm The 3 rectangles are similar. Find the unknown sides p and q 1. Comparing corresponding sides in A and C. SF = 35.5/5 = x7.1. 2. Apply the scale factor to find the unknown side. q = 7.1 x 2.1 = 14.91 cm
- 12. 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 Two triangles are similar if their •Corresponding angles are equal
- 13. 70o 70o 45o 65o 45o These two triangles are similar since they are equiangular. 50o 55o 75o 50o 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
- 14. Finding Unknown Sides 20 cm 15 cm 12 cm 6 cm b c Since the triangles are equiangular they are similar. So comparing corresponding sides to find the scale factor of enlargement. SF = 15/12 = x1.25. b = 1.25 x 6 = 7.5 cm c = 20/1.25 = 16 cm
- 15. 31.5 cm 14 cm 8 cm 6 cm x y SF = 14/8 = x1.75. x = 1.75 x 6 = 10.5 cm y = 31.5/1.75 = 18 cm Since the triangles are equiangular they are similar. So comparing corresponding sides to find the scale factor of enlargement. Finding Unknown Sides
- 16. Determining similarity A B E D Triangles ABC and DEC are similar. Why? C Angle ACB = angle ECD (Vertically Opposite) Angle ABC = angle DEC (Alt angles) Angle BAC = angle EDC (Alt angles) Since ABC is similar to DEC we know that corresponding sides are in proportion ABï‚®DE BCï‚®EC ACï‚®DC The order of the lettering is important in order to show which pairs of sides correspond.
- 17. A B C D E If BC is parallel to DE, explain why triangles ABC and ADE are similar Angle BAC = angle DAE (common to both triangles) Angle ABC = angle ADE (corresponding angles between parallels) Angle ACB = angle AED (corresponding angles between parallels) A D E A D E B C B C A line drawn parallel to any side of a triangle produces 2 similar triangles. Triangles EBC and EAD are similar Triangles DBC and DAE are similar
- 18. A tree 5m high casts a shadow 8 m long. Find the height of a tree casting a shadow 28 m long. Example Problem 1 5m 8m 28m h Explain why the triangles must be similar.   28 3.5 8 sf  1 3 7 .5 5 5 . h x m
- 19. A B C D E 20m 45m 5m y The two triangles below are similar: Find the distance y.   50 10 5 sf    20 10 2 m y Example Problem 2
- 20. A B C D E In the diagram below BE is parallel to CD and all measurements are as shown. (a) Calculate the length CD (b) Calculate the perimeter of the Trapezium EBCD 4.8 m 6 m 3 m 4.2 m 9 m A C D 7.2m 7.2m 2.1 m 6.3m   9 ( ) 6 1.5 f a s    1.5 4.8 7.2 CD x m So AC = 4.2 6 3 .5 1 . x m  BC 6.3 - 4.2 = 2.1 m  Perimeter = 7.2 + 3 + 4.8 + 2.1 17 = .1 m Example Problem 3