Section 8-2 Similar Polygons SPI 31A: identify corresponding parts of congruent geometric figures SPI 32C: determine congruence or relations between triangles or quadrilaterals given a diagram SPI 22B: apply ratio and proportion to solve real-world problems Objectives: identify and determine missing measurements of polygons apply proportion to the real-world Similar (~): two figures have the same shape but not necessarily the same size. Figures are ~ if: (1) corresponding s are (2) corresponding sides are proportional Similarity Ratio: ratio of the lengths of corresponding sides
Understanding Similarity ABCD ~ EFGH AB EF = AD ? Since the two figures are similar: What is the m E?m E = m A = 53º Complete the statement for corresponding sides: EH
Determine Similarity Determine whether the triangles are similar. If they are write a similarity statement and give the similarity ratio. Conditions for similarity: 1. corresponding s are 2. corresponding sides are proportional AC FD = = AB FE = = BC ED = = ∆ ABC ~ ∆ FED with a similarity ratio of ¾ or 3:4
Real-World Connection: Drawing You want to draw an enlargement of a design that is painted on a 3in by 5in card. You plan to draw on a 8 ½ in by 11 in piece of paper. What are the dimensions of the largest complete enlargement you can draw? 3 in 5 in = x 11 Postcard Paper Width Length Write and solve a proportion 5x = 33 x = 6.6 The largest complete enlargement you can draw is 6.6 in by 11 in.
Golden Rectangle and Phi Ø The golden rectangle is considered pleasing to the human eye. It is used extensively in architecture and art. In any golden rectangle, the length and the width are in the golden ratio which is about : 1. Total 13 units 8 units 5 units Golden ratio = 13 units Golden ratio = 8 Make a square
Bottom and top of the oval: Bottom to naval: 12 units Top to naval: 7.5 units Golden ratio: 12:7.5 or 1.6 Bottom of figure to top of head Bottom to naval: 9.5 units Top to naval: 5.8 units Golden ratio: 9.5 : 5.8 or 1.64 The golden rectangle and golden ratio depicted in Art