6.1 Ratios, Proportions, and the Geometric Mean


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1 6.1 Ratios, Proportions, and the Geometric Mean Obj.: Solve problems by writing and solving proportions. Key Vocabulary Ratio  If a and b are two numbers or quantities and b 0, then the ratio of a to b is a/b. The ratio of a to b can also be written as a: b. Proportion  An equation that states that two ratios are equal is called a proportion. Means, extremes  The numbers b and c are the means of the proportion. The numbers a and d are the extremes of the proportion. KEY CONCEPT A Property of Proportions 1. Cross Products Property In a proportion, the product of the extremes equals the product of the means. If b a = d c where b 0 and d 0, then a d = b c. 2 4 = 3 6, 2 6 = 3 4, 12 = 12 Geometric mean  The geometric mean of two positive numbers a and b is the a x positive number x that satisfies x = b. So, x² = ab and x = ab. EXAMPLE 1 Simplify ratios Simplify the ratio. (See Table of Measures, p. 921) a. 76 cm: 8 cm b. 4 ft 24in. EXAMPLE 2 Use a ratio to find a dimension PAINTING You are painting barn doors. You know that the perimeter of the wall is 64 feet and that the ratio of its length to its height is 3:5. Find the area of the wall.
2 EXAMPLE 3 Use extended ratios ALGEBRA The measures of the angles in BCD are in the extended ratio of 2: 3: 4. Find the measures of the angles. EXAMPLE 4 Solve proportions ALGEBRA Solve the proportion. 3 x a. 4 = 16 b. x 3 1 = 2 x EXAMPLE 5 Solve a realworld problem Bowling You want to find the total number of rows of boards that make up 24 lanes at a bowling alley. You know that there are 117 rows in 3 lanes. Find the total number of rows of boards that make up the 24 lanes. EXAMPLE 6 Find a geometric mean Find the geometric mean of 16 and 48.
3 6.1 Cont.
4 6.2 Use Proportions to Solve Geometry Problems Obj.: Use proportions to solve geometry problems. Key Vocabulary Scale drawing  A scale drawing is a drawing that is the same shape as the object it represents. Scale  The scale is a ratio that describes how the dimensions in the drawing are related to the actual dimensions of the object. KEY CONCEPT Additional Properties of Proportions 2. Reciprocal Property If two ratios are equal, then their reciprocals are also equal. If b a = d c, then a b = c d 3. If you interchange the means of a proportion, then you form another true proportion. If b a = d c, then c a = d b 4. In a proportion, if you add the value of each ratio s denominator to its numerator, then you form another true proportion. a If b c a =, then d b b = c d d EXAMPLE 1 Use properties of proportions In the diagram, AC = BC. Write four true proportions. DF EF EXAMPLE 2 Use proportions with geometric figures ALGEBRA In the diagram, JL JK. Find JH and JL. LH = KG
5 EXAMPLE 3 Find the scale of a drawing Keys The length of the scale drawing is 7 centimeters. The length of the actual key is 4 centimeters. What is the scale of the drawing? EXAMPLE 4 Use a scale drawing MAPS The scale of the map at the right is 1 inch: 8 miles. Find the actual distance from Westbrook to Cooley. EXAMPLE 5 TAKS Reasoning: MultiStep Problem SCALE MODEL You buy a 3D scale model of the Sunsphere in Knoxville, TN. The actual building is 266 feet tall. Your model is 20 inches tall, and the diameter of the dome on your scale model is about 5.6 inches. a. What is the diameter of the actual dome? b. About how many times as tall as your model is the actual building?
6 6.2 Cont.
7 6.3 Use Similar Polygons Obj.: Use proportions to identify similar polygons. Key Vocabulary Similar polygons  Two polygons are similar polygons if corresponding angles are congruent and corresponding side lengths are proportional. Scale factor  If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the scale factor. ABCD EFGH Order Matters!! Corresponding angles A E, B F, C G, and D H Ratios of corresponding sides AB EF Perimeters of Similar Polygons Theorem If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths. KL LM MN NK If KLMN PQRS, then = KL = LM PQ QR RS SP PQ QR = BC FG = CD GH = MN = NK. RS SP DA = HE KEY CONCEPT Corresponding Lengths in Similar Polygons If two polygons are similar, then the ratio of any two corresponding lengths in the polygons is equal to the scale factor of the similar polygons. EXAMPLE 1 Use similarity statements In the diagram, ABC DEF. a. List all pairs of congruent angles. b. Check that the ratios of corresponding side lengths are equal. c. Write the ratios of the corresponding side lengths in a statement of proportionality.
8 EXAMPLE 2 Find the scale factor Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor of ABCD to JKLM. (6.3 cont.) EXAMPLE 3 Use similar polygons ALGEBRA In the diagram, BCD RST. Find the value of x. EXAMPLE 4 Find perimeters of similar figures Basketball A large cement court is being poured for a basketball hoop in place of a smaller one. The court will be 20 feet wide and 25 feet long. The old court was similar in shape, but only 16 feet wide. a. Find the scale factor of the new court to the old court. b. Find the perimeter of the new court and the old court. EXAMPLE 5 Use a scale factor In the diagram, TPR XPZ. Find the length of the altitude GL.
9 6.3 Cont.
10 6.4 Prove Triangles Similar by AA Obj.: Use the AA Similarity Postulate. Key Vocabulary Similar polygons  Two polygons are similar polygons if corresponding angles are congruent and corresponding side lengths are proportional. AngleAngle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. JKL ~ XYZ EXAMPLE 1 Use the AA Similarity Postulate Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning. EXAMPLE 2 Show that triangles are similar Show that the two triangles are similar. a. RTV and RQS b. LMN and NOP
11 EXAMPLE 3 Using similar triangles Height A lifeguard is standing beside the lifeguard chair on a beach. The lifeguard is 6 feet 4 inches tall and casts a shadow that is 48 inches long. The chair casts a shadow that is 6 feet long. How tall is the chair?
12 6.5 Prove Triangles Similar by SSS and SAS Obj.: Use the SSS and SAS Similarity Theorems. Key Vocabulary ratio, p. 356 proportion, p. 358 similar polygons, p. 372 SideSideSide (SSS) Similarity Theorem If the corresponding side lengths of two triangles are proportional, then the triangles are similar. If = =, then ABC ~ RST. SideAngleSide (SAS) Similarity Theorem If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. If X M and =, then XYZ ~ MNP EXAMPLE 1 Use the SSS Similarity Theorem Is either DEF or GHJ similar to ABC? EXAMPLE 2 Use the SSS Similarity Theorem ALGEBRA Find the value of x that makes ABC ~ DEF.
13
14 6.6 Use Proportionality Theorems Obj.: Use proportions with a triangle or parallel lines. Key Vocabulary Corresponding angles  Two angles are corresponding angles if they have corresponding positions. For example, 2 and 6 are above the lines and to the right of the transversal t. Ratio  If a and b are two numbers or quantities and b 0, then the ratio of a to b is a/b. The ratio of a to b can also be written as a : b. Proportion  An equation that states that two ratios are equal is called a proportion. Triangle Proportionality Theorem prop. Th If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. If, then = Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. If =, then 3 lines inters. 2 trans. prop. If three parallel lines intersect two transversals, then they divide the transversals proportionally. ray bis. of opp. side prop. to other 2 sides 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 1 Find the length of a segment In the diagram,, RQ = 10, RS = 12, and ST = 6, What is the length of? =
15 EXAMPLE 2 Solve a realworld problem Aerodynamics A spoiler for a remote controlled car is shown where AB = 31 mm, BC = 19, CD = 27 mm, and DE = 23 mm. Explain why is not parallel to. EXAMPLE 3 Use Theorem 6.6 Farming A farmer s land is divided by a newly constructed interstate. The distances shown are in meters. Find the distance CA between the north border and the south border of the farmer s land. EXAMPLE 4 Use Theorem 6.7 In the diagram, DEG GEF. Use the given side lengths to find the length of.
16 6.6 Cont.
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