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Transcript
Honors Geometry Chapter 6: Proportions and Similarity Objective: 6.1 Proportions -write ratios -use properties of proportions ratio A ratio is a comparison of two quantities. The ratio a to b, where b is not a zero, can be written as or a:b. b Examples a. In 2002, the Chicago Cubs baseball team won 67 games out of 162. Write a ratio for the number of games won to the total number of games played. b. A doll house that is 15 inches tall is a scale model of a real house with a height of 20 feet. What is the ratio of the height of the doll house to the height of the real house? proportion A statement that two ratios are equal is called a proportion. Solve for x. Examples a. 6x 4 27 3 b. x2 8 3 9 c. x2 x4 4 2 d. Find the measures of the angles in a triangle where the ratio of the measures of the angles is 4:5:6. Assignment 6.1 page 285 #18-35 1 6.2 Similar Polygons (Agilemind: Topic 14) Objective: -Identify similar figures -solve problems involving scale factors Consider the table below. Analyze the pairs of shapes in the Similar column and the pairs of shapes in the Not Similar column. Then, think of a definition for similarity based on your observations. similar polygons Example Is polygon WXYZ polygon PQRS? Scale factor 2 Example If ABCD EFGH, find each of the following. 1. scale factor of ABCD to EFGH = 2. EF = 3. FG = 5. perimeter of ABCD = 4. GH = 6. perimeter of EFGH = 7. ratio of perimeter of ABCD to perimeter of EFGH = Assignment 6.2 Page 293 #11-20, 27-33, 35, 37, 39 3 6.3 Similar Triangles Objective: -Identify similar triangles -use similar triangles to solve problems There are three ways to determine whether two triangles are similar. AA Similarity • Show that two angles of one triangle are congruent to two angles of the other triangle. (AA Similarity) SSS Similarity • Show that the measures of the corresponding sides of the triangles are proportional. (SSS Similarity) SAS Similarity • Show that the measure of two sides of a triangle are proportional to the measures of the corresponding sides of the other triangle and that the included angles are congruent. (SAS Similarity) Examples Determine whether each pair of triangles is similar. Give a reason for your answer. 1. 4 Identify the similar triangles in each figure. Explain why they are similar and find the missing measures. 2. 3. Find x and y 4. Find x and y Indirect Example: Measurement A lighthouse casts a 128-foot shadow. A nearby lamppost that measures 5 feet 3 inches casts an 8-foot shadow. a. Write a proportion that can be used to determine the height of the lighthouse. b. What is the height of the lighthouse? Assignment 6.3 page 302 #10-21 all 32, 41,42 5 6.4 Parallel Lines and Parts (Agilemind: Topic 15) Objective: -use proportional parts of a triangle -divide a segment into parts Triangle If a line is parallel to one side of a triangle and intersects the other two sides in Proportionality two distinct points, then it separates these sides into segments of proportional Theorem lengths Converse of the Triangle If a line intersects two sides of a triangle and separates the sides into Proportionality corresponding segments of proportional lengths, then the line is parallel to the Theorem third side. Midsegment A segment whose endpoints are the midpoints of two sides of the triangle Triangle Midsegment a midsegment is parallel to the third side and is half its length. Theorem Corollary 6.1 If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. Corollary 6.2 If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. 6 Example In ▲ABC, EF || CB. Find x. Example Find x and y Example Triangle ABC has vertices A(-2,2), B(2,4), and C(4,-4). DE is a midsegment of ▲ABC. a. Find the coordinates of D and E. b. Verify that BC DE. c. Verify that DE = ½ BC. Assignment 6.4 Part I: page 312 #1525 odd Assignment 6.4 Part II: page 313 27, 29, 30, 33, 34 7 6.5 Parts of Similar Triangles Objective: -recognize and use proportional relationships of corresponding perimeters of similar triangles -recognize and use proportional relationships of corresponding angle bisectors, altitudes, and medians of similar triangles Proportional If two triangles are similar, then the perimeters are proportional to the measures of Perimeters Theorem the corresponding sides. Example Each pair of triangles is similar. Find the perimeter of the indicated triangle. Special Segments of Similar Triangles a. If two triangles are similar, then the measures of the corresponding altitudes are proportional to the measures of the corresponding sides. b. If two triangles are similar, then the measures of the corresponding angle bisectors of the triangles are proportional to the measures of the corresponding sides. c. If two triangles are similar, then the measures of the corresponding medians are proportional to the measures of the corresponding sides. 8 Example Find x for each pair of similar triangles. a. b. c. Angle Bisector Theorem An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides. Example Find x for each pair of similar triangles. Assignment 6.5 page 319 #3-7, 17, 21, 23, 25, 26 9 10