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Transcript
Sec. 8.1/8.2 Ratios and Proportions ratio – comparison of two numbers usually in fraction form where the denominator is not zero a/b a to b a:b proportion – two ratios that are set equal to each other geometric mean – In the following proportion x is the geometric mean of a and b. Properties of Proportions 1. The product of the means equals the product of the extremes. 2. If ratios are equal, then their reciprocals are equal. 3. If a c b d , then a b c d 4. If a c b d , then ab cd b d Examples : 1. Simplify 12cm 4m 2. Simplify 6 ft 8in True or False p r 3. If 6 10 , then 4. If a c , 3 4 then p 3 r 5 . a3 c3 3 4 . 5. The perimeter of a rectangle is 60 cm. The ratio of AB to BC is 3 : 2. Find the length and the width of the rectangle. 6. The measures of the angles of a triangle is 1 : 2 : 3, find the measures of the angles. 4 5 7. Solve the proportion : x 7 8. Solve the proportion : 3 2 y2 y 9. 5. A scale model of the Titanic is 107.5 inches long by 11.25 inches wide. The Titanic was 882.75 feet long…how wide was it? Sec. 8.3 Similar Polygons Similar polygons – polygons with corresponding angles congruent and corresponding sides proportional. scale factor – the ratio of the corresponding sides What is the scale factor for the similar polygons? What is the ratio of their perimeters? Thm. 8.1 If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths. Example : What is the scale factor of the similar polygons? Find z. Sec. 8.4 Similar Triangles similar triangles – triangles that have congruent, corresponding angles and proportional, corresponding sides. Postulate 25 Angle-Angle(AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Example : 1. In the diagram, ∆BTW~∆ETC. a. Write the statement of proportionality. b. Find mTEC c. Find ET and BE 2. Find the length of the altitude segment QS. Sec. 8.5 Proving Triangles Are Similar Side-Side-Side (SSS) Similarity Theorem If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar. Side-Angle-Side (SAS) Similarity Theorem If the angle of one triangle is congruent to an angle of another triangle and the sides that include these angles are proportional, then the triangles are similar. Examples : 1. Which two triangles are similar? 2. What is the distance across the river? Sec. 8.6 Proportions and Similar Triangles Thm. 8.4 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. Thm. 8.5 Converse of the Triangle Prop. Thm. If a line divides two sides of the triangle proportionally, then it is parallel to the third side. Thm. 8.6 If three parallel lines intersect two transversals, then they divide the transversals proportionally. Thm. 8.7 If a ray bisects an angle of triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. Sec. 8.7 Dilations Dilations – a non-rigid transformation in which the preimage is similar to the image. The following properties are true : Reduction : 0 < k < 1 Enlargement : k > 1