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Chapter 10: Similarity BY: JUSTIN KIM & KEVIN PRAETORIUS Lesson 1: Ratio and Proportion The ratio of the number a to the number b is the number . A proportion is an equality between ratios A proportion can be represented symbolically as = a is the first term of a proportion b is the second c is the third d is the fourth Lesson 1 Continued The 2nd and 3rd terms are the means 1st and 4th terms are the extremes The product of the means equals the product of the extremes If you take = , you can cross multiply to get ad=cb If the means are equal, they are a geometric mean The number b is the geometric mean between the numbers a and c if = Lesson 2: Similar Figures Two triangles are similar iff there is a correspondence between their vertices such that their corresponding sides are proportional and their corresponding angles are equal. The center of a dilation is the point at which a shape is dilated The magnitude of a dilation is the relative size of an image compared with the original Lesson 3: The Side Splitter Theorem The Side Splitter Theorem- If a line parallel to one side of a triangle intersects the other two sides in different points, it divides the sides in the same ratio. Corollary- If a line parallel to one side of a triangle intersects the other two sides in different points, it cuts off segments proportional to the sides. The Side Splitter Theorem A XZ is || to BC therefore it is a side splitter to ABC… X Therefore Z Or… = = Which is the Corollary B C Lesson 4: The AA similarity Theorem The Angle Angle (AA) Theorem- If two angles of one triangle are equal to two angles of another triangle, the triangles are similar. Corollary- Two triangles similar to a third triangle are similar to each other AA similarity D A ABC ~ DEF A If B C B E F C A ~ B and B ~ C, then A ~ C Lesson 5: Proportions and Dilations Corresponding altitudes of triangles are altitudes that are drawn from corresponding vertices. Corresponding altitudes of similar triangles have the same ratio as that of the E corresponding sides. B A C D F Lesson 6: Perimeters and Areas of Similar Figures The ratio of the perimeters of two similar polygons is equal to the ratio of their corresponding sides. The ratio of the areas of two similar polygons is equal to the square of the ratio of their corresponding sides. So, if the ratio of the sides of two similar triangles is , the ratio of their perimeters is and the ratios of their areas is Additional Lesson: The Angle Bisector Theorem The angle bisector theorem states that an angle bisector in a triangle divides the opposite side into segments that have the same ratio as the other two sides. As a proportion, B X = A C Extra Homework Problems In the extra homework problems we had to use the Side Splitter Theorem, AA Similarity Theorem, and Angle Bisector Theorem to find the areas and perimeters of triangles. One of the problems asks you to find length x given that the line drawn is an angle bisector of the triangle. Extra Homework Problems First, set up a proportion. = Next, cross multiple the proportion. 36 = 10x - 30 Now simplify and solve for x. 66 = 10x 6.6 = x