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Similarity Unit Essential Question: How are the ratios of similar figures used to find missing parts? Essential Question How do you use ratios and proportions to solve problems? Ratios & Proportions A ratio is a comparison between two quantities: a:b a to b What is a ratio? a/b How big is the actual bedroom? Scale Drawings Does the poster need to be cropped? A proportion is a statement that two ratios are equal: 15 10 45 30 An extended proportion: 36 18 2 48 24 3 Proportions Properties Examples These trains are all similar: What is SIMILARITY? Essential Question: When are polygons similar? Similar Polygons Two polygons are SIMILAR (~) if: ◦ Corresponding Angles are CONGRUENT ◦ Corresponding Sides are PROPORTIONAL The ratio between the sides is called the SIMILARITY RATIO ABCD ~ EFGH ◦ m<B = ◦ m<E = Similarity Determine if the triangles are similar. If so, write the similarity statement and the similarity ratio. Example LMNO ~ QRST Find x Find SR Using Similar Triangles A Golden Rectangle is a rectangle that can be split into a Square and a Similar Rectangle: In a Golden Rectangle, the ratio of the length to the width is the Golden Ratio, approximately 1.618:1 Golden Rectangle Sufficient Conditions? Essential Question: How do you prove triangles are similar? Proving Triangles Similar What are sufficient conditions for Similarity? If two angles are congruent, then... AA~ Postulate Prove the triangles are similar: Using AA~ SAS~ Create an extended proportion of 3 sides. SSS~ Show that the triangles are similar. Write a similarity statement. Example Prove these triangles are similar: Proving ~ Show the triangles are similar. Then find DE. Applying ~ Using INDIRECT MEASUREMENT How tall is the cactus? If m<1 = 65°, then find the measure of every numbered angle: What can you conclude about the three triangles? Consider this… Essential Question: What special ratios can be formed using segments within triangles? Similarity in Right Triangles Explanation: Altitude to Hypotenuse Proportions with the same number in the numerator and denominator form geometric means: 3 x x 48 x 3 48 2 So, How is this similar to Arithmetic Means (averages)? Geometric Means Find the Geometric Means of: 4 and 16 5 and 45 Finding Geometric Means Applying the Corollaries Small, Medium, Large Opener Essential Question: What special ratios can be formed using segments within triangles? Proportions in Triangles For example: Side-Splitter Theorem Similar Triangles: Compare: Side-Splitter: Corollary to Side-Splitter: Sailing Triangle-Angle-Bisector Examples: Essential Question: How does a change in the linear dimension of a figure affect the perimeter, circumference, and area of a figure? Perimeters & Areas of Similar Polygons Perimeter & Area Ratios Similarity Ratio: Perimeter Ratio: Area Ratio: Perimeter & Area Ratios Similar Figures Real-World Finding Ratios