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Chapter 8 Similarity 8.1 Ratio and Proportion Ratios Ratio- Comparison of 2 quantities in the same units The ratio of a to b can be written as a/b a : b The denominator cannot be zero Simplifying Ratios Ratios should be expressed in simplified form 6:8 = 3:4 Before reducing, make sure that the units are the same. 12in : 3 ft 12in : 36 in 1: 3 Examples (page 461) Simplify each ratio 10. 16 students 24 students 12. 22 feet 52 feet 18. 60 cm 1m Examples (page 461) Simplify each ratio 20. 2 mi 3000 ft 24. 20 oz. 4 lb There are 5280 ft in 1 mi. There are 16 oz in 1 lb. Examples (page 461) Find the width to length ratio 14. 16 mm 20 mm 16. 12 in. 2 ft Using Ratios Example 1 The perimeter of the isosceles triangle shown is 56 in. The ratio of LM : MN is 5:4. Find the length of the sides and the base of the triangle. L N M Using Ratios Example 2 The measures of the angles in a triangle are in the extended ratio 3:4:8. Find the measures of the angles 4x 8x 3x Using Ratios Example 3 The ratios of the side lengths of ΔQRS to the corresponding side lengths of ΔVTU are 3:2. Find the unknown lengths. Q U T 2 cm V S 18 cm R Proportions Proportion Ratio = Ratio Fraction = Fraction Means and Extremes Extreme: Mean = Mean: Extreme Extreme Mean Mean Extreme Solving Proportions Solving Proportions Cross multiply Let the means equal the extremes Example: 3 5 x 20 Properties of Proportions Cross Product Property a c If , then ad bc b d Reciprocal Property a c b d If , then b d a c Solving Proportions Example 1 9 6 14 x Solving Proportions Example 2 s5 s 4 10 Solving Proportions Example 3 A photo of a building has the measurements shown. The actual building is 26 ¼ ft wide. How tall is it? 2.75 in 1 7/8 in 8.2 Problem solving in Geometry with Proportions Properties of Proportions a c a b If , then b d c d a c ab cd If , then b d b d Example 1 Tell whether the statement is true or false A. s 15 s 3 If , then 10 t t 2 B. 3 5 3 x 5 y If , then x y x y Example 2 In the diagram MQ LQ MN LP Find the length of LQ. M 6 N 15 13 Q L 5 P Geometric Mean Geometric Mean The geometric mean between two numbers a and b is the positive number x such that a x x b ex: 8/4 = 4/2 Example 3 Find the geometric mean between 4 and 9. Similar Polygons Polygons are similar if and only if the corresponding angles are congruent and the corresponding sides are proportionate. Similar figures are dilations of each other. (They are reduced or enlarged by a scale factor.) The symbol for similar is Example 1 Determine if the sides of the polygon are proportionate. 8m 12 m 6m 8m 6m Example 2 Determine if the sides of the polygon are proportionate. 15 m 9m 5m 3m 12 m 4m Example 3 Find the missing measurements. HAPIE NWYRS 6 A H 18 P 5 AP = EI = SN = YR = E I 4 W 24 Y N S 21 R Example 4 Find the missing measurements. QUAD SIML A D S 12 L 65º 8 125º Q 20 25 I 95º U M QD = MI = mD = mU = mA = 8.4/8.5 Similar Triangles Similar Triangles To be similar, corresponding sides must be proportional and corresponding angles are congruent. Similarity Shortcuts AA Similarity Shortcut If two angles in one triangle are congruent to two angles in another triangle, then the triangles are similar. Similarity Shortcuts SSS Similarity Shortcut If three sides in one triangle are proportional to the three sides in another triangle, then the triangles are similar. Similarity Shortcuts SAS Similarity Shortcut If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. Similarity Shortcuts We have three shortcuts: AA SAS SSS Example 1 9 g 6 4 7 10.5 Example 2 k 32 h 50 24 30 Example 3 36 42 m 24 4. A flagpole 4 meters tall casts a 6 meter shadow. At the same time of day, a nearby building casts a 24 meter shadow. How tall is the building? 4 m 6m 24m 5. Five foot tall Melody casts an 84 inch shadow. How tall is her friend if, at the same time of day, his shadow is 1 foot shorter than hers? 6. A 10 meter rope from the top of a flagpole reaches to the end of the flagpole’s 6 meter shadow. How tall is the nearby football goalpost if, at the same moment, it has a shadow of 4 meters? 10m 6m 4m 7. Private eye Samantha Diamond places a mirror on the ground between herself and an apartment building and stands so that when she looks into the mirror, she sees into a window. The mirror is 1.22 meters from her feet and 7.32 meters from the base of the building. Sam’s eye is 1.82 meters above the ground. How high is the window? 1.82 1.22 7.32 8.6 Proportions and Similar Triangles Proportions Using similar triangles missing sides can be found by setting up proportions. Theorem Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. Q T RT RU If TU || QS ,then . TQ US R S U Theorem Converse of the Triangle Proportionality Theorem Q If a line divides two sides of a triangle proportionally, then it is parallel to the third side. RT RU T If TQ R S U US , thenTU || QS . Example 1 In the diagram, segment UY is parallel to segment VX, UV = 3, UW = 18 and XW = 16. What is the length of segment YX? U V W Y X Example 2 Given the diagram, determine whether segment PQ is parallel to segment TR. Q 9.75 P R 9 26 T 24 S Theorem If three parallel lines intersect two transversals, then they divide the transversals proportionally. Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. Example 3 In the diagram, 1 2 3, AB =6, BC=9, EF=8. What is x? C 9 B 6 A 1 D 3 2 8 x E F Example 4 In the diagram, LKM MKN. Use the given side lengths to find the length of segment MN. 15 L N M 3 17 K 5. Juanita, who is 1.82 meters tall, wants to find the height of a tree in her backyard. From the tree’s base, she walks 12.20 meters along the tree’s shadow to a position where the end of her shadow exactly overlaps the end of the tree’s shadow. She is now 6.10 meters from the end of the shadows. How tall is the tree? 6.10 1.82 12.20 8.7 Dilations C 3 P 9 P’ Dilations Dilation: Transformation that maps all points so that the proportion CP ' stands CP true. Enlargement: A dilation which makes the transformed image larger than the original image Reduction: A dilation which makes the transformed image smaller than the original image. Enlargement An enlargement has a scale factor of k which if found by the proportion CP . ' In an enlargement k is always greater than 1. CP Find k: C 3 P 9 P’ Reduction A reduction has a scale factor of k CP ' which is found by the proportion . In CP a reduction, 0 < k < 1. C P Find k 6 P’ 14 Dilations in a coordinate plane If the center of the dilation is the origin, the image can be found by multiplying each coordinate by the scale factor Example: Original coordinates: (3, 6), (6, 12) and (9, 3) Scale factor: 1/3 Find the image coordinates.