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8.1 Ratio and Proportion Ratio of a to b: the quotient a/b if a and b are two quantities that are measured in the same units. It can also be written as a:b. Ratios are generally expressed in simplified form. 6:8 is simplified to 3:4 Simplify the ratios. 60 cm 1m 8 in 1 ft 2 lb 8 oz 2640 ft 1 mile 1 kg 800 g Using ratios The perimeter of the isosceles triangle is 56 in. The ratio of LM:MN is 5:4. Find the measure of each side of the triangle. L M N The measures of the angles in a triangle are in the ratio of 3:4:8. Find the measures of the angles. Proportion: an equation that equates two ratios. Cross Product Property: the product of the means = the product of the extremes. If a = c , then ad = bc a and d are the extremes b d b and c are the means Reciprocal Property: if two ratios are equal, then their reciprocals are also equal. If a = c , then b = d b d a c Solving Proportions 4=x 2 = 13 12 9 3 m x+2=x 3 4 8 = 14 t 5 m – 5 = 25 4 10 The ratio of gravity of Venus to Earth is 9:10. If your math teacher weighs 200 pounds on Earth, how much would he weigh on Venus? 461: 1 – 19, 21 – 22, 25 – 50 8.2 Problem Solving in Geometry with Proportions Additional Properties: if a = c, then a = b b d c d if a = c, then a + b = c + d b d b c Using Properties of Proportions In the diagram AB = AC. Find the length of BD. BD CE A D B C E Geometric Mean: for two positive numbers a and b, the positive number x such that a = x or x = ab. x b Find the geometric mean of the two numbers. 8 and 18 4 and 25 6 and 8 9 and 36 Using Proportions in Real Life A scale model of the Titanic is 107.5 inches long and 11.25 inches wide. The Titanic itself was 882.75 feet long. How wide was it? Length of model = Width of model Length of Titanic Width of Titanic Or Length of model = Length of Titanic Width of model Width of Titanic 468: 1 – 20, 23 – 28, 31 – 34 8.3 Similar Polygons Similar Polygons: two polygons such that their corresponding angles are congruent and the lengths of corresponding sides are proportional. The symbol is ~. A B Q R ABCD ~ QRST D C T S List all the pairs of congruent angles and write the ratios of the corresponding sides: <A = < , <B = < , <C = < , <D = < , AB = BC = CD = DA Decide whether the figures are similar. If so, write a similarity statement. A 15 H 9 6 10 B C J 8 K 12 8 You have a picture that measures 3.5 inches by 5 inches and you want to enlarge it to be 16 inches wide. How long should it be? Theorem: If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths. If KLMN ~ PQRS, then KL + LM + MN + NK = KL = LM = MN = NK PQ + QR + RS + SP PQ QR RS SP Day One: 475: 1 – 7, 8, 10, 11 – 38 all 8.4 Similar Triangles In the diagram, GED ~ GFH. Write the statement of proportionality. Find m<GFH and m<GHF Find GF and FE AA Similarity Postulate: if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. If <A = <D and <B = <E, then ABC ~ DEF Examples: In the diagram LMN ~ PQN m<M = ____, m<P = ____ MN = ____, QM = ____ Write a statement of proportionality: ------- = ------ = -----Example 4: page 482: Use the proportion: f = n, to solve if g = 50, f = 8, and n = 3. h g Use the same proportion and solve if f = 10, n = 6, and g = 100. Day One: 483: 1 – 8, 10 – 28 all 8.5 Proving Triangles are Similar In the previous section we learned we can prove triangles are similar using the Angle-Angle Similarity Postulate (AA). Today we find two more ways to prove triangles are similar. SSS: Side-Side-Side Theorem: If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar. If AB = BC = CA DE EF FD then ABC ~ DEF SAS: Side-Angle-Side Theorem: If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the two triangles are similar. If AB = BC and m<B = m<E, then ABC ~ DE EF DEF Which of the following triangles are similar? To determine the answer, compare short, middle, and long sides. Repeat the previous example with triangles with the following measures: Triangle One: 6, 15, 18; Triangle Two: 3, 5, 7; Triangle Three: 10, 25, 35 In the figure AC = 6, AD = 10, BC = 9, and BE = 15. How can you prove the triangles are similar? A E C B D Examples 5 and 6 on page 491: 492: 1 – 29 all: you need rulers and protractors 8.6 Proportions and Similar Triangles 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. If QS ll TU, then RT = RU TQ US Converse of Triangle Proportionality Theorem: if a line divides two sides of a triangle proportionally, then it is parallel to the third side. If RT = RU , then QS ll TU TQ US If three parallel lines intersect two transversals, then they divide the transversals proportionally. If r ll s and s ll t and l and m intersect r, s, and t, then UW = VX. WY XZ If a ray bisects an angle of a triangle, then it divides the opposite sides into segments whose lengths are proportional to the lengths of the other sides. If CD bisects <ACB, then AD = CA DB CB Examples 1, 2, 3, and 4 on pages 498 – 500. 502: 1 – 30, 36, 37, 39 all