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6-3: Use Similar Polygons Objectives: 1. To define and use similar polygons Assignment: • P. 367-9: 11, 13, 14, 22, 23 • P. 376-8: 1-3, 6, 8-13, 19, 20, 31, 32, 36, 39, 40-42 • Challenge Problems Objective 1 You will be able to define and use similar polygons Guggenheim Museum Building big stuff can be expensive. So to work out details, artists and architects usually build scale models. (Frank Lloyd Wright is a giant) Guggenheim Museum Building big stuff can be expensive. So to work out details, artists and architects usually build scale models. Guggenheim Museum A scale model is similar to the actually object that is to be built. And that does not mean that they are kind of alike. Guggenheim Museum A scale model is similar to the actually object that is to be built. And that does not mean that they are kind of alike. Similarity Figures that have the same shape but not necessarily the same size are similar figures. But what does “same shape mean”? Are Mr. Noel’s heads similar? Similarity Similar shapes can be thought of as enlargements or reductions with no irregular distortions. So two shapes are similar if one can be enlarged or reduced so that it is congruent to the original. Investigation 1 Use the “What are Similar Polygons” handout to discover a mathematical definition of similar polygons. The two illustrations represent the enlargement and reduction of an “impossible” 3-D solid. Similar Polygons Two polygons are similar polygons iff the corresponding angles are congruent and the corresponding sides are proportional. Similarity Statement: N C πΆππ π~ππ΄πΌπ Corresponding Angles: N C ∠πΆ ≅ ∠π ∠π ≅ ∠π΄ ∠π ≅ ∠πΌ ∠π ≅ ∠π O M Z R A O Statement of Proportionality: R πΆπ ππ π π ππΆ = = = ππ΄ π΄πΌ πΌπ ππ A I Solving a Proportion What’s the relationship between the cross products of a proportion? 2.4ο150 360 ο½ 2.4 150 1 They’re equal! 360ο1 Solving a Proportion Cross Products Property In a proportion, the product of the extremes equals the product of the means. Solving a Proportion To solve a proportion involving a variable, simply set the two cross products equal to each other. Then solve! 275 ο½ 25 15β25 275βx 15 x 375ο½ 275x 1.36ο½ x Example 1 Solve the proportion. 26 x ο½ 50 75 Example 2 Solve the proportion. 2 x ο«1 ο½ xο«4 5 More Proportion Properties Example 3 Use the definition of similar polygons to find the measure of x and y, assuming SMAL ~ BIGE. Example 2 When asked to find the length of π·πΈ given that the triangles are similar, Kenny says 10. Explain what is wrong with Kenny’s reasoning. D A 6 5 F C 3 8 B E 10 Scale Factor C In similar polygons, the ratio of two corresponding sides is called a scale factor. N 8 4 5 O 6 R Z 12 M 6 4 6 Scale factor = = 2 3 7.5 A 9 I Corresponding Lengths Corresponding Lengths in Similar Polygons If two polygons are similar, then the ratio of any two corresponding lengths in the polygons is equal to the scale factor of the similar polygons. Sides Altitudes Medians Midsegments Example 4 In the diagram Δπππ ~Δπππ. Find the length of the altitude ππ. Investigation 2 Similar polygons preserve angle measures and create proportional lengths, but what about their perimeters and areas? Similarity Relationships For two shapes with a scale factor of π: π, each of the following relationships will be true: Perimeter Linear Units π: π Area Square Units π2 : π2 Volume Cubic Units π3 : π3 Example 5 In the diagram, ABCDE ~ FGHJK. Find the perimeter of ABCDE. F 15 G 9 A 10 H B C 18 12 E D K 15 J Example 6 In the diagram, ABCDE ~ FGHJK. Find the area of ABCDE. 6-3: Use Similar Polygons Objectives: 1. To define and use similar polygons Assignment: • P. 367-9: 11, 13, 14, 22, 23 • P. 376-8: 1-3, 6, 8-13, 19, 20, 31, 32, 36, 39, 40-42 • Challenge Problems