Geometry. Chapter 6 Resource Masters

Transcription

1 Geometry hapter 6 esource asters

2 6 eading to Learn athematics Vocabulary uilder his is an alphabetical list of the key vocabulary terms you will learn in hapter 6. s you study the chapter, complete each term s definition or description. emember to add the page number where you found the term. dd these pages to your Geometry tudy Notebook to review vocabulary at the end of the chapter. cross products Vocabulary erm Found on age efinition/escription/ample Vocabulary uilder etremes fractal iteration I uh shuhn means (continued on the net page) Glencoe/cGraw-Hill vii Glencoe Geometry

3 6 eading to Learn athematics Vocabulary uilder (continued) midsegment Vocabulary erm Found on age efinition/escription/ample proportion ratio scale factor self-similar similar polygons Glencoe/cGraw-Hill viii Glencoe Geometry

4 6 Learning to ead athematics roof uilder his is a list of key theorems and postulates you will learn in hapter 6. s you study the chapter, write each theorem or postulate in your own words. Include illustrations as appropriate. emember to include the page number where you found the theorem or postulate. dd this page to your Geometry tudy Notebook so you can review the theorems and postulates at the end of the chapter. heorem or ostulate heorem 6.1 ide-ide-ide () imilarity Found on age escription/illustration/bbreviation roof uilder heorem 6.2 ide-ngle-ide () imilarity heorem 6.3 heorem 6.4 riangle roportionality heorem heorem 6.5 onverse of the riangle roportionality heorem heorem 6.6 riangle idsegment heorem (continued on the net page) Glencoe/cGraw-Hill i Glencoe Geometry

5 6 Learning to ead athematics roof uilder (continued) heorem or ostulate heorem 6.7 roportional erimeters heorem Found on age escription/illustration/bbreviation heorem 6.8 heorem 6. heorem 6.10 heorem 6.11 ngle isector heorem ostulate 6.1 ngle-ngle () imilarity Glencoe/cGraw-Hill Glencoe Geometry

6 6-1 tudy Guide and Intervention roportions Write atios ratio is a comparison of two quantities. he ratio a to b, where b is not zero, can be written as a b or a:b. he ratio of two quantities is sometimes called a scale factor. For a scale factor, the units for each quantity are the same. ample 1 In 2002, the hicago ubs baseball team won 67 games out of 162. Write a ratio for the number of games won to the total number of games played. o find the ratio, divide the number of games won by the total number of games played. he 67 result is, which is about he hicago ubs won about 41% of their games in ample 2 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? o start, convert the height of the real house to inches. 20 feet 12 inches per foot 240 inches o find the ratio or scale factor of the heights, divide the height of the doll house by the height of the real house. he ratio is 15 inches:240 inches or 1:16. he height of the doll 1 house is the height of the real house. 1 6 Lesson 6-1 ercises 1. In the 2002 ajor League baseball season, ammy osa hit 4 home runs and was at bat 556 times. Find the ratio of home runs to the number of times he was at bat. 2. here are 182 girls in the sophomore class of 305 students. Find the ratio of girls to total students. 3. he length of a rectangle is 8 inches and its width is 5 inches. Find the ratio of length to width. 4. he sides of a triangle are 3 inches, 4 inches, and 5 inches. Find the scale factor between the longest and the shortest sides. 5. he length of a model train is 18 inches. It is a scale model of a train that is 48 feet long. Find the scale factor. Glencoe/cGraw-Hill 25 Glencoe Geometry

7 6-1 Use roperties of roportions statement that two ratios are equal is called a proportion. In the proportion a b c, where b and d are not zero, the values a and d are d the etremes and the values b and c are the means. In a proportion, the product of the means is equal to the product of the etremes, so ad bc. a b c d a d b c etremes tudy Guide and Intervention (continued) roportions means ample 1 olve ross products 432 ultiply. 48 ivide each side by. ample 2 room is 4 centimeters by 28 centimeters on a scale drawing of a house. For the actual room, the larger dimension is 14 feet. Find the shorter dimension of the actual room. If is the room s shorter dimension, then ross products 8 ivide each side by 4. he shorter side of the room is 8 feet. ercises shorter dimension longer dimension olve each proportion y y Use a proportion to solve each problem. 7. If 3 cassettes cost $44.85, find the cost of one cassette. 8. he ratio of the sides of a triangle are 8:15:17. If the perimeter of the triangle is 480 inches, find the length of each side of the triangle.. he scale on a map indicates that one inch equals 4 miles. If two towns are 3.5 inches apart on the map, what is the actual distance between the towns? Glencoe/cGraw-Hill 26 Glencoe Geometry

8 6-1 kills ractice roportions 1. FOOLL tight end scored 6 touchdowns in 14 games. Find the ratio of touchdowns per game. 2. UION In a schedule of 6 classes, arta has 2 elective classes. What is the ratio of elective to non-elective classes in arta s schedule? 3. IOLOG Out of 274 listed species of birds in the United tates, 78 species made the endangered list. Find the ratio of endangered species of birds to listed species in the United tates. 4. n artist in ortland, Oregon, makes bronze sculptures of dogs. he ratio of the height of a sculpture to the actual height of the dog is 2:3. If the height of the sculpture is 14 inches, find the height of the dog. Lesson HOOL he ratio of male students to female students in the drama club at ampbell High chool is 3:4. If the number of male students in the club is 18, what is the number of female students? olve each proportion Find the measures of the sides of each triangle. 12. he ratio of the measures of the sides of a triangle is 3:5:7, and its perimeter is 450 centimeters. 13. he ratio of the measures of the sides of a triangle is 5:6:, and its perimeter is 220 meters. 14. he ratio of the measures of the sides of a triangle is 4:6:8, and its perimeter is 126 feet. 15. he ratio of the measures of the sides of a triangle is 5:7:8, and its perimeter is 40 inches. Glencoe/cGraw-Hill 27 Glencoe Geometry

9 6-1 ractice roportions 1. NUIION One ounce of cheddar cheese contains grams of fat. i of the grams of fat are saturated fats. Find the ratio of saturated fats to total fat in an ounce of cheese. 2. FING he ratio of goats to sheep at a university research farm is 4:7. he number of sheep at the farm is 28. What is the number of goats? 3. dward Hopper s oil on canvas painting Nighthawks has a length of 60 inches and a width of 30 inches. print of the original has a length of 2.5 inches. What is the width of the print? olve each proportion Find the measures of the sides of each triangle. 10. he ratio of the measures of the sides of a triangle is 3:4:6, and its perimeter is 104 feet. 11. he ratio of the measures of the sides of a triangle is 7::12, and its perimeter is 84 inches. 12. he ratio of the measures of the sides of a triangle is 6:7:, and its perimeter is 77 centimeters. Find the measures of the angles in each triangle. 13. he ratio of the measures of the angles is 4:5: he ratio of the measures of the angles is 5:7: IG he span of the enjamin Franklin suspension bridge in hiladelphia, ennsylvania, is 1750 feet. model of the bridge has a span of 42 inches. What is the ratio of the span of the model to the span of the actual enjamin Franklin ridge? Glencoe/cGraw-Hill 28 Glencoe Geometry

10 6-1 eading to Learn athematics roportions re-ctivity How do artists use ratios? eading the Lesson ead the introduction to Lesson 6-1 at the top of page 282 in your tetbook. stimate the ratio of length to width for the background rectangles in iffany s lematis kylight. 1. atch each description in the first column with a word or phrase from the second column. a. he ratio of two corresponding quantities i. proportion b. r and u in the equation r s t u ii. cross products c. a comparison of two quantities iii. means d. ru and st in the equation r s t u iv. scale factor e. an equation stating that two ratios are equal v. etremes f. s and t in the equation r s t u vi. ratio Lesson If m, n, p, and q are nonzero numbers such that m n p, which of the following q statements could be false?. np mq. n p m q. mp nq. qm pn. m n q p q n F. p m G. m:p n:q H. m:n p:q 3. Write two proportions that match each description. a. eans are 5 and 8; etremes are 4 and 10. b. eans are 5 and 4; etremes are positive integers that are different from means. Helping ou emember 4. ometimes it is easier to remember a mathematical idea if you put it into words without using any mathematical symbols. How can you use this approach to remember the concept of equality of cross products? Glencoe/cGraw-Hill 2 Glencoe Geometry

11 6-1 nrichment Golden ectangles Use a straightedge, compass, and the instructions below to construct a golden rectangle. 1. onstruct square with sides of 2 cm. 2. onstruct the midpoint of. all the midpoint. 3. raw. et your compass at an opening equal to. Use as the center to draw an arc that intersects. all the point of intersection. 4. onstruct a line through that is perpendicular to. 5. raw so that it intersects the perpendicular line in step 4. all the intersection point Q. Q is a golden rectangle because the ratio of its length to its width is heck this conclusion by finding the value of Q. ectangles whose sides have this ratio are, it is said, the most pleasing to the human eye. Q figure consisting of similar golden rectangles is shown below. Use a compass and the instructions below to draw quarter-circle arcs that form a spiral like that found in the shell of a chambered nautilus. 6. Using as a center, draw an arc that passes through and. 7. Using as a center, draw an arc that passes through and. 8. Using F as a center, draw an arc that passes through and G.. Using H as a center, draw an arc that passes through G and J. J L F N H K G 10. Using K as a center, draw an arc that passes through J and L. 11. Using as a center, draw an arc that passes through L and N. Glencoe/cGraw-Hill 300 Glencoe Geometry

12 6-2 tudy Guide and Intervention imilar olygons Identify imilar Figures ample 1 etermine whether the triangles are similar. wo polygons are similar if and only if their corresponding angles are congruent and their corresponding sides are proportional. Z because they are right angles, and X. y the hird ngle heorem,. For the sides, 2 0, , and 2 0 XZ 23 X 23 Z he side lengths are proportional. o XZ X Z ample 2 Is polygon WXZ polygon Q? For the sides, W X Q 8 2, X Q 12 2, Z , W 3. o corresponding sides are proportional. 2 and Z 6 lso, W, X Q,, and Z, so corresponding angles are congruent. We can conclude that polygon WXZ polygon Q. 12 W Z 15 X 18 8 Q Lesson 6-2 ercises etermine whether each pair of figures is similar. If they are similar, give the ratio of corresponding sides equilateral triangles y z y 12z Glencoe/cGraw-Hill 301 Glencoe Geometry

13 6-2 cale Factors When two polygons are similar, the ratio of the lengths of corresponding sides is called the scale factor. t the right, XZ. he scale factor of to XZ is 2 and the scale factor of XZ to is 1 2. tudy Guide and Intervention (continued) imilar olygons he two polygons are similar. Find and y. ample 1 ample N y Use the congruent angles to write the corresponding vertices in order. N Write proportions to find and y y (13) 32y 38(16) 26 y 1 ercises 10 cm 8 cm 6 cm 4 cm 5 cm Z X 3 cm. Find the scale factor and find the lengths of and. (0, 0) ach pair of polygons is similar. Find and y. y (1, 3) (3, 0) (, 0) and 3 6. he scale factor of to is 6:3 or 2:1. Using the istance Formula, 1 10 and he lengths of the sides of are twice those of, so 2() or 210 and 2() or y y y y In ample 2 above, point has coordinates (5, 6). Use the istance Formula to verify the lengths of and. Glencoe/cGraw-Hill 302 Glencoe Geometry

14 6-2 kills ractice imilar olygons etermine whether each pair of figures is similar. Justify your answer F 3 3 Q 3 3 Z 7.5 W X 7.5 ach pair of polygons is similar. Write a similarity statement, and find, the measure(s) of the indicated side(s), and the scale factor. 3. GH 4. and U H 6 F 13 G 4 W 3 4 X 5 5 U Lesson W 6. and 5 Q U V W 8 L 10 N 2 1 Glencoe/cGraw-Hill 303 Glencoe Geometry

15 6-2 ractice imilar olygons etermine whether each pair of figures is similar. Justify your answer L K J Q U V ach pair of polygons is similar. Write a similarity statement, and find, the measure(s) of the indicated side(s), and the scale factor. 3. L and N 4. and F 14 N 6 10 L F OOIN GO riangle has vertices (0, 0), (4, 0), and (2, 4). he coordinates of each verte are multiplied by 3 to create F. how that F is similar to. 6. INIO IGN Graham used the scale drawing of his living room to decide where to place furniture. Find the dimensions of the living room if the scale in the drawing is 1 inch 4.5 feet in. 4 in. Glencoe/cGraw-Hill 304 Glencoe Geometry

16 6-2 eading to Learn athematics imilar olygons re-ctivity How do artists use geometric patterns? ead the introduction to Lesson 6-2 at the top of page 28 in your tetbook. escribe the figures that have similar shapes. eading the Lesson What happens to the figures as your eyes move from the center to the outer edge? 1. omplete each sentence. a. wo polygons that have eactly the same shape, but not necessarily the same size, are. b. wo polygons are congruent if they have eactly the same shape and the same. c. wo polygons are similar if their corresponding angles are and their corresponding sides are. d. wo polygons are congruent if their corresponding angles are and their corresponding sides are. e. he ratio of the lengths of corresponding sides of two similar figures is called the. f. ultiplying the coordinates of all points of a figure in the coordinate plane by a scale factor to get a similar figure is called a. g. If two polygons are similar with a scale factor of 1, then the polygons are. Lesson etermine whether each statement is always, sometimes, or never true. a. wo similar triangles are congruent. b. wo equilateral triangles are congruent. c. n equilateral triangle is similar to a scalene triangle. d. wo rectangles are similar. e. wo isosceles right triangles are congruent. f. wo isosceles right triangles are similar. g. square is similar to an equilateral triangle. h. wo acute triangles are similar. i. wo rectangles in which the length is twice the width are similar. j. wo congruent polygons are similar. Helping ou emember 3. good way to remember a new mathematical vocabulary term is to relate it to words used in everyday life. he word scale has many meanings in nglish. Give three phrases that include the word scale in a way that is related to proportions. Glencoe/cGraw-Hill 305 Glencoe Geometry

17 6-2 nrichment onstructing imilar olygons Here are four steps for constructing a polygon that is similar to and with sides twice as long as those of an eisting polygon. tep 1 hoose any point either inside or outside the polygon and label it O. tep 2 tep 3 tep 4 raw rays from O through each verte of the polygon. For verte V, set the compass to length OV. hen locate a new point V on ray OV such that VV OV. hus, OV 2(OV ). epeat tep 3 for each verte. onnect points V, W, X and to form the new polygon. wo constructions of polygons similar to and with sides twice those of VWX are shown below. Notice that the placement of point O does not affect the size or shape of VWX, only its location. V V W W V W V W V W O O X X X X X race each polygon. hen construct a similar polygon with sides twice as long as those of the given polygon F G 3. plain how to construct a similar polygon with sides three times the length of those of polygon HIJKL. hen do the construction. H I 4. plain how to construct a similar polygon times the length of those of polygon NQ. hen do the construction. N L K J Q Glencoe/cGraw-Hill 306 Glencoe Geometry

18 6-3 tudy Guide and Intervention imilar riangles Identify imilar riangles Here are three ways to show that two triangles are similar. imilarity imilarity imilarity wo angles of one triangle are congruent to two angles of another triangle. he measures of the corresponding sides of two triangles are proportional. he measures of two sides of one triangle are proportional to the measures of two corresponding sides of another triangle, and the included angles are congruent. etermine whether the triangles are similar. ample 1 ample F F F by imilarity. ercises F etermine whether the triangles are similar N 6 Q , so N N. Q mn m, so N. N Q by imilarity. etermine whether each pair of triangles is similar. Justify your answer Lesson Glencoe/cGraw-Hill 307 Glencoe Geometry

19 6-3 tudy Guide and Intervention (continued) imilar riangles Use imilar riangles imilar triangles can be used to find measurements. Find and y. ample 1 ample 2 F F 183 y F F F y (183) y y 36 person 6 feet tall casts a 1.5-foot-long shadow at the same time that a flagpole casts a 7-foot-long shadow. How tall is the flagpole? 1.5 ft 6 ft 7 ft? he sun s rays form similar triangles. Using for the height of the pole, 6 1.5, 7 so and 28. he flagpole is 28 feet tall. ercises ach pair of triangles is similar. Find and y y y y y y y he heights of two vertical posts are 2 meters and 0.45 meter. When the shorter post casts a shadow that is 0.85 meter long, what is the length of the longer post s shadow to the nearest hundredth? Glencoe/cGraw-Hill 308 Glencoe Geometry

20 6-3 kills ractice imilar riangles etermine whether each pair of triangles is similar. Justify your answer W X Q U J K Q 30 LG Identify the similar triangles, and find and the measures of the indicated sides. 5. and 6. JL and L J K L 3 N 4 Lesson H and F 8. U and F H V G U Glencoe/cGraw-Hill 30 Glencoe Geometry

21 6-3 ractice imilar riangles etermine whether each pair of triangles is similar. Justify your answer. 1. J W K L N LG Identify the similar triangles, and find and the measures of the indicated sides. 3. L and Q 4. NL and L L 3 18 N 12 Q 1 J N 5 8 K L Use the given information to find each measure. 5. If Q, 6, 7, 6. If F HI, F 3, G 1, Q 8, and 1, HI 4, and HG 3, find and. find G and HG. Q F G I H INI UN For ercises 7 and 8, use the following information. lighthouse casts a 128-foot shadow. nearby lamppost that measures 5 feet 3 inches casts an 8-foot shadow. 7. Write a proportion that can be used to determine the height of the lighthouse. 8. What is the height of the lighthouse? Glencoe/cGraw-Hill 310 Glencoe Geometry

22 6-3 eading to Learn athematics imilar riangles re-ctivity How do engineers use geometry? ead the introduction to Lesson 6-3 at the top of page 28 in your tetbook. What does it mean to say that triangular shapes result in rigid construction? What would happen if the shapes used in the construction were quadrilaterals? eading the Lesson 1. tate whether each condition guarantees that two triangles are congruent or similar. If the condition guarantees that the triangles are both similar and congruent, write congruent. If there is not enough information to guarantee that the triangles will be congruent or similar, write neither. a. wo sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle. b. he measures of all three pairs of corresponding sides are proportional. c. wo angles of one triangle are congruent to two angles of the other triangle. d. wo angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of the other triangle. e. he measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle, and the included angles are congruent. f. he three sides of one triangle are congruent to the three sides of the other triangle. g. he three angles of one triangle are congruent to the three angles of the other triangle. h. One acute angle of a right triangle is congruent to one acute angle of another right triangle. i. he measures of two sides of a triangle are proportional to the measures of two sides of another triangle. Lesson Identify each of the following as an eample of a refleive, symmetric, or transitive property. a. If UVW, then UVW. b. If UVW and UVW OQ, then OQ. c. Helping ou emember 3. good way to remember something is to eplain it to someone else. uppose one of your classmates is having trouble understanding the difference between for congruent triangles and for similar triangles. How can you eplain the difference to him? Glencoe/cGraw-Hill 311 Glencoe Geometry

23 6-3 nrichment atio uzzles with riangles If you know the perimeter of a triangle and the ratios of the sides, you can find the lengths of the sides. ample he perimeter of a triangle is 84 units. he sides have lengths r, s, and t. he ratio of s to r is 5:3, and the ratio of t to r is 2:1. Find the length of each side. ince both ratios contain r, rewrite one or both ratios to make r the same. ou can write the ratio of t to r as 6:3. Now you can write a three-part ratio. r: s : t 3:5:6 here is a number such that r 3, s 5, and t 6. ince you know the perimeter, 84, you can use algebra to find the lengths of the sides. r s t , 5 30, 6 36 o r 18, s 30, and t 36. Find the lengths of the sides of each triangle. 1. he perimeter of a triangle is 75 units. he sides have lengths a, b, and c. he ratio of b to a is 3:5, and the ratio of c to a is 7:5. Find the length of each side. 2. he perimeter of a triangle is 88 units. he sides have lengths d, e, and f. he ratio of e to d is 3:1, and the ratio of f to e is 10:. Find the length of each side. 3. he perimeter of a triangle is 1 units. he sides have lengths p, q, and r. he ratio of p to r is 3:1, and the ratio of q to r is 5:2. Find the length of each side. 4. he perimeter of a triangle is 68 units. he sides have lengths g, h, and j. he ratio of j to g is 2:1, and the ratio of h to g is 5:4. Find the length of each side. 5. Write a problem similar to those above involving ratios in triangles. Glencoe/cGraw-Hill 312 Glencoe Geometry

24 6-4 tudy Guide and Intervention arallel Lines and roportional arts roportional arts of riangles In any triangle, a line parallel to one side of a triangle separates the other two sides proportionally. he converse is also true. If X and are the midpoints of and, then X is a midsegment of the triangle. he riangle idsegment heorem states that a midsegment is parallel to the third side and is half its length. If X X,then X. If X,then X. X If X is a midsegment, then X 1 and X 2. X ample 1 ample F 2 F ince F, F In, F. Find. triangle has vertices (3, 6), (3, 2), and F(7, 2). idsegment GH is parallel to F. Find the length of GH. GH is a midsegment, so its length is onehalf that of F. oints and F have the same y-coordinate, so F 7 (3) 10. he length of midsegment GH is 5. ercises Find Lesson In ample 2, find the slope of F and show that F GH. Glencoe/cGraw-Hill 313 Glencoe Geometry

25 6-4 tudy Guide and Intervention (continued) arallel Lines and roportional arts ivide egments roportionally When three or more parallel lines cut two transversals, they separate the transversals into proportional parts. If the ratio of the parts is 1, then the parallel lines separate the transversals into congruent parts. t a b s 1 c d 2 3 u v n w m If 1 2 3, then a b c. d If and u v 1, then w 1. ample so 3 8 efer to lines 1, 2,and 3 above. If a 3, b 8, and c 5, find d hen 3d 40 and d 13 d 3. ercises Find and y y 3 1 2y 2 5 y 8 y y 32 y 3 Glencoe/cGraw-Hill 314 Glencoe Geometry

26 6-4 kills ractice arallel Lines and roportional arts 1. If JK 7, KH 21, and JL 6, 2. Find and V if U 8, U 14, find LI. V 1 and V K J H L U V I etermine whether , 12, 10, and 8 4., 27, and is one third of 5. 30, 45, and is twice OOIN GO For ercises 6 8, use the following information. riangle has vertices (5, 2), (1, 8), and (4, 2). oint is the midpoint of and is the midpoint of. y 6. Identify the coordinates of and. O 7. how that is parallel to. 8. how that 1 2. Lesson 6-4. Find and y. 10. Find and y y 8 7 y y 1 3y 5 Glencoe/cGraw-Hill 315 Glencoe Geometry

27 If 24, 27, and 18, 2. Find, Q, and if Q 6, find. 12, 27, and 4. ractice arallel Lines and roportional arts Q etermine whether JK N. 3. JN 18, JL 30, K 21, and L 35 J K N L 4. K 24, KL 44, and NL 5 6 JN OOIN GO For ercises 5 and 6, use the following information. riangle FG has vertices (4, 1), F(2, 5), and G(2, 1). oint K is the midpoint of G and H is the midpoint of FG. 5. how that F is parallel to KH. K O y F H G 6. how that KH 1 2 F. 7. Find and y. 8. Find and y y y y 2 4y he distance from Wilmington to sh Grove along Kendall is 820 feet and along agnolia, 660 feet. If the distance between eech and sh Grove along agnolia is 280 feet, what is the distance between the two streets along Kendall? agnolia Wilmington eech sh Grove Kendall Glencoe/cGraw-Hill 316 Glencoe Geometry

28 6-4 eading to Learn athematics arallel Lines and roportional arts re-ctivity How do city planners use geometry? ead the introduction to Lesson 6-4 at the top of page 307 in your tetbook. Use a geometric idea to eplain why the distance between hicago venue and Ontario treet is shorter along ichigan venue than along Lake hore rive. eading the Lesson 1. rovide the missing words to complete the statement of each theorem. hen state the name of the theorem. a. If a line intersects two sides of a triangle and separates the sides into corresponding segments of lengths, then the line is to the third side. b. midsegment of a triangle is to one side of the triangle and its length is the length of that side. c. If a line is to one side of a triangle and intersects the other two sides in distinct points, then it separates these sides into of proportional length. 2. efer to the figure at the right. a. Name the three midsegments of. b. If 8, U 3, and W 5, find the length of each of the midsegments. c. What is the perimeter of? d. What is the perimeter of UVW? e. What are the perimeters of UV, VW, and UW? f. How are the perimeters of each of the four small triangles related to the perimeter of the large triangle? g. Would the relationship that you found in part f apply to any triangle in which the midpoints of the three sides are connected? U V W Lesson 6-4 Helping ou emember 3. good way to remember a new mathematical term is to relate it to other mathematical vocabulary that you already know. What is an easy way to remember the definition of midsegment using other geometric terms? Glencoe/cGraw-Hill 317 Glencoe Geometry

29 6-4 nrichment arallel Lines and ongruent arts here is a theorem stating that if three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on any transversal. his can be shown for any number of parallel lines. he following drafting technique uses this fact to divide a segment into congruent parts. is to be separated into five congruent parts. his can be done very accurately without using a ruler. ll that is needed is a compass and a piece of notebook paper. tep 1 Hold the corner of a piece of notebook paper at point. tep 2 From point, draw a segment along the paper that is five spaces long. ark where the lines of the notebook paper meet the segment. Label the fifth point,. tep 3 raw. hrough each of the other marks on, construct a line parallel to. he points where these lines intersect will divide into five congruent segments. Use a compass and a piece of notebook paper to divide each segment into the given number of congruent parts. 1. si congruent parts 2. seven congruent parts Glencoe/cGraw-Hill 318 Glencoe Geometry

30 6-5 tudy Guide and Intervention arts of imilar riangles erimeters If two triangles are similar, their perimeters have the same proportion as the corresponding sides. If, then. ample Use the diagram above with. If 24 and 15, find the ratio of their perimeters. ince, the ratio of the perimeters of and is the same as the ratio of corresponding sides. perimeter of herefore 2 4 perimeter of ercises ach pair of triangles is similar. Find the perimeter of the indicated triangle. 1. XZ X Z XZ N Z X Lesson 6-5 Glencoe/cGraw-Hill 31 Glencoe Geometry

31 6-5 tudy Guide and Intervention (continued) arts of imilar riangles pecial egments of imilar riangles When two triangles are similar, corresponding altitudes, angle bisectors, and medians are proportional to the corresponding sides. lso, in any triangle an angle bisector separates the opposite side into segments that have the same ratio as the other two sides of the triangle. In the figure, XZ, with angle bisectors as shown. Find. ample 1 ample ince XZ, the measures of the angle bisectors are proportional to the measures of a pair of corresponding sides. X W (8) ercises X 8 W Find for each pair of similar triangles. Z U 20 U bisects. Find. ince U is an angle bisector, U. U (15) Glencoe/cGraw-Hill 320 Glencoe Geometry

32 6-5 kills ractice arts of imilar riangles Find the perimeter of the given triangle. 1. JKL, if JKL, 14, 2. F, if F, 27, 12, 10, and LJ 14 16, 25, and F 15 J L K F 3. Q, if Q LN, L 16, 4. KL, if KL FGH, FG 30, N 14, NL 27, and 18 GH 38, HF 38, and KL 24 L N Q H G L K F Use the given information to find each measure. 5. Find FG if FG, H is an 6. Find N if N, is an altitude of, FJ is an altitude of altitude of, Q is an altitude of FG, 6, H 5, and FJ 7. N, 24, 14, and Q F H J G Q N Find. 7. HKL XZ 8. K 10 H 18 L X 15 Z Lesson 6-5 Glencoe/cGraw-Hill 321 Glencoe Geometry

33 6-5 ractice arts of imilar riangles Find the perimeter of the given triangle. 1. F, if F, 36, 2. U, if U KL, KL 12, 20, 40, and 35 L 31, K 32, and U 28 U L F K Use the given information to find each measure. 3. Find if JKL N, K is an 4. Find Z if U XZ, U is an altitude of JKL, is an altitude of altitude of U, Z is an altitude of N, KL 28, K 18, and XZ, U 8.5, U 6, and Z K Z U J L N X Find HOOGH For ercises 7 and 8, use the following information. Francine has a camera in which the distance from the lens to the film is 24 millimeters. 7. If Francine takes a full-length photograph of her friend from a distance of 3 meters and the height of her friend is 140 centimeters, what will be the height of the image on the film? (Hint: onvert to the same unit of measure.) 8. uppose the height of the image on the film of her friend is 15 millimeters. If Francine took a full-length shot, what was the distance between the camera and her friend? Glencoe/cGraw-Hill 322 Glencoe Geometry

34 6-5 eading to Learn athematics arts of imilar riangles re-ctivity How is geometry related to photography? ead the introduction to Lesson 6-5 at the top of page 316 in your tetbook. How is similarity involved in the process of making a photographic print from a negative? Why do photographers place their cameras on tripods? eading the Lesson 1. In the figure, UVW. omplete each proportion involving the lengths of segments in this figure by replacing the question mark. hen identify the definition or theorem from the list below that the completed proportion illustrates. X U Q W N V i. efinition of congruent polygons ii. efinition of similar polygons iii. roportional erimeters heorem iv. ngle isectors heorem v. imilar triangles have corresponding altitudes proportional to corresponding sides. vi. imilar triangles have corresponding medians proportional to corresponding sides. vii. imilar triangles have corresponding angle bisectors proportional to corresponding sides. a.? U V c. UN? VW b. U W?X d. UV? e.? f. U N UW? g. W Q? h. U W? Q VW V Helping ou emember 2. good way to remember a large amount of information is to remember key words. What key words will help you remember the features of similar triangles that are proportional to the lengths of the corresponding sides? Lesson 6-5 Glencoe/cGraw-Hill 323 Glencoe Geometry

35 6-5 nrichment roportions for imilar riangles ecall that if a line crosses two sides of a triangle and is parallel to the third side, then the line separates the two sides that it crosses into segments of proportional lengths. ou can write many proportions by identifying similar triangles in the following diagram. In the diagram, N, FL, and Q. F G H J K L N Q nswer each question. Use the diagram above. 1. Name a triangle similar to GN. 2. Name a triangle similar to JH. 3. Name two triangles similar to JK. 4. Name a triangle similar to. omplete each proportion. 5. G F? 8. H G? 6. H??. J L J? 7. J J L 10. N? olve. 11. If J 16, J 48, and L 30, find L. 12. If K 5, K 7, and J 8, find J. 13. If N 12, N 32, and 48, find G. ound to the nearest tenth. 14. If H 18, H 82, and 130, find J. 15. Write three more problems that can be solved using the diagram above. Glencoe/cGraw-Hill 324 Glencoe Geometry

36 6-6 tudy Guide and Intervention Fractals and elf-imilarity haracteristics of Fractals he act of repeating a process over and over, such as finding a third of a segment, then a third of the new segment, and so on, is called iteration. When the process of iteration is applied to some geometric figures, the results are called fractals. For objects such as fractals, when a portion of the object has the same shape or characteristics as the entire object, the object can be called self-similar. ample In the diagram at the right, notice that the details at each stage are similar to the details at tage 1. Lesson 6-6 tage 1 tage 2 tage 3 ercises 1. Follow the iteration process below to produce a fractal. tage 1 raw a square. raw an isosceles right triangle on the top side of the square. Use the side of the square as the hypotenuse of the triangle. raw a square on each leg of the right triangle. tage 2 epeat the steps in tage 1, drawing an isosceles triangle and two small squares for each of the small squares from tage 1. tage 3 epeat the steps in tage 1 for each of the smallest squares in tage Is the figure produced in tage 3 self-similar? Glencoe/cGraw-Hill 325 Glencoe Geometry

37 6-6 Nongeometric Iteration n iterative process can be applied to an algebraic epression or equation. he result is called a recursive formula. ample tudy Guide and Intervention (continued) Fractals and elf-imilarity Find the value of 3, where the initial value of is 2. epeat the process three times and describe the pattern. Initial value: 2 First time: econd time: hird time: ,217,728 he result of each step of the iteration is used for the net step. For this eample, the values are greater with each iteration. here is no maimum value, so the values are described as approaching infinity. ercises For ercises 1 5, find the value of each epression. hen use that value as the net in the epression. epeat the process three times, and describe your observations. 1., where initially equals , where initially equals , where initially equals where initially equals , where initially equals Harpesh paid $1000 for a savings certificate. It earns interest at an annual rate of 2.8%, and interest is added to the certificate each year. What will the certificate be worth after four years? Glencoe/cGraw-Hill 326 Glencoe Geometry

38 6-6 kills ractice Fractals and elf-imilarity tages 1 and 2 of a fractal known as the antor set are shown. o get tage 2, the segment in tage 1 is trisected, and the interior of the middle segment is removed. (he interior of a segment is the segment with its endpoints removed.) he process is repeated for subsequent stages. tage 1 tage 2 Lesson raw stages 3 and 4 of the antor set. tage 3 tage 4 2. How many segments are there in tage 3? tage 4? 3. What happens to the length of the line segments in each stage? 4. he antor set is the set of points after infinitely many iterations. Is the antor set self-similar? Find the value of each epression. hen, use that value as the net in the epression. epeat the process three times, and describe your observations , where initially equals , where initially equals 1 Find the first three iterates of each epression , where initially equals , where initially equals 8 Glencoe/cGraw-Hill 327 Glencoe Geometry

39 6-6 n artist is designing a book cover and wants to show a copy of the book cover in the lower left corner of the cover. fter tages 1 and 2 of the design, he realizes that the design is developing into a fractal! tage 1 ractice Fractals and elf-imilarity H tage 2 H H 1. raw tages 3 and 4 of the book cover fractal. tage 3 tage 4 H H H H H H H 2. fter infinitely many iterations, will the result be a self similar fractal? 3. On what part of the book cover should you focus your attention to be sure you can find a copy of the entire figure? Find the value of each epression. hen, use that value as the net in the epression. epeat the process three times and describe your observations. 4. 2( 3), where initially equals , where initially equals 10 Find the first three iterates of each epression , where initially equals , where initially equals 2 8. HOUING he ndrews purchased a house for $6,000. he real estate agent who sold the house said that comparable houses in the area appreciate at a rate of 4.5% per year. If this pattern continues, what will be the value of the house in three years? ound to the nearest whole number. Glencoe/cGraw-Hill 328 Glencoe Geometry

40 6-6 eading to Learn athematics Fractals and elf-imilarity re-ctivity How is mathematics found in nature? ead the introduction to Lesson 6-6 at the top of page 325 in your tetbook. Name two objects from nature other than broccoli in which a small piece resembles the whole. Lesson 6-6 eading the Lesson 1. atch each definition from the first column with a term from the second column. (ome words of phrases in the second column may be used more than once or not at all.) hrase erm a. a geometric figure that is created using iteration i. similar b. a pattern in which smaller and smaller details of a shape have ii. iteration the same geometric characteristics as the original shape iii. fractal c. the result of translating an iterative process into a formula or iv. self-similar algebraic equation v. recursion d. the process of repeating the same procedure over and over formula again vi. congruent 2. efer to the three patterns below. i. ii. iii. a. Give the special name for each of the fractals you obtain by continuing the pattern without end. b. Which of these fractals are self-similar? c. Which of these fractals are strictly self-similar? Helping ou emember 3. good way to remember a new mathematical term is to relate it to everyday nglish words. he word fractal is related to fraction and fragment. Use your dictionary to find at least one definition for each of these words that you think is related to the meaning of the word fractal and eplain the connection. Glencoe/cGraw-Hill 32 Glencoe Geometry

41 6-6 nrichment he öbius trip öbius strip is a special surface with only one side. It was discovered by ugust Ferdinand öbius, a German astronomer and mathematician. 1. o make a öbius strip, cut a strip of paper about 16 inches long and 1 inch wide. ark the ends with the letters,,, and as shown below. wist the paper once, connecting to and to.ape the ends together on both sides. 2. Use a crayon or pencil to shade one side of the paper. hade around the strip until you get back to where you started. What happens? 3. What do you think will happen if you cut the öbius strip down the middle? ry it. 4. ake another öbius strip. tarting a third of the way in from one edge, cut around the strip, staying always the same distance in from the edge. What happens? 5. tart with another long strip of paper. wist the paper twice and connect the ends. What happens when you cut down the center of this strip? 6. tart with another long strip of paper. wist the paper three times and connect the ends. What happens when you cut down the center of this strip? Glencoe/cGraw-Hill 330 Glencoe Geometry

42 nswers (Lesson 6-1) Lesson tudy Guide and Intervention roportions Write atios ratio is a comparison of two quantities. he ratio a to b, where b is not zero, can be written as a or a:b. he ratio of two quantities is sometimes called a scale b factor. For a scale factor, the units for each quantity are the same. ample 1 In 2002, the hicago ubs baseball team won 67 games out of 162. Write a ratio for the number of games won to the total number of games played. o find the ratio, divide the number of games won by the total number of games played. he 6 7 result is, which is about he hicago ubs won about 41% of their games in ample 2 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? o start, convert the height of the real house to inches. 20 feet 12 inches per foot 240 inches o find the ratio or scale factor of the heights, divide the height of the doll house by the height of the real house. he ratio is 15 inches:240 inches or 1:16. he height of the doll 1 house is 1 the height of the real house. 6 ercises 1. In the 2002 ajor League baseball season, ammy osa hit 4 home runs and was at bat 556 times. Find the ratio of home runs to the number of times he was at bat here are 182 girls in the sophomore class of 305 students. Find the ratio of girls to total students he length of a rectangle is 8 inches and its width is 5 inches. Find the ratio of length to width he sides of a triangle are 3 inches, 4 inches, and 5 inches. Find the scale factor between the longest and the shortest sides he length of a model train is 18 inches. It is a scale model of a train that is 48 feet long. Find the scale factor Glencoe/cGraw-Hill 25 Glencoe Geometry 6-1 Use roperties of roportions statement that two ratios are equal is called a proportion. In the proportion a b c, where b and d are not zero, the values a and d are d the etremes and the values b and c are the means. In a proportion, the product of the means is equal to the product of the etremes, so ad bc. a b c d a d b c etremes means tudy Guide and Intervention (continued) roportions ample 1 1 olve ross products ultiply. 48 ivide each side by. 7. ample 2 room is 4 centimeters by 28 centimeters on a scale drawing of a house. For the actual room, the larger dimension is 14 feet. Find the shorter dimension of the actual room. If is the room s shorter dimension, then shorter dimension longer dimension ross products 8 ivide each side by 4. he shorter side of the room is 8 feet. ercises olve each proportion y y Use a proportion to solve each problem. 7. If 3 cassettes cost $44.85, find the cost of one cassette. $ he ratio of the sides of a triangle are 8:15:17. If the perimeter of the triangle is 480 inches, find the length of each side of the triangle. 6 in., 180 in., 204 in.. he scale on a map indicates that one inch equals 4 miles. If two towns are 3.5 inches apart on the map, what is the actual distance between the towns? 14 mi Glencoe/cGraw-Hill 26 Glencoe Geometry Glencoe/cGraw-Hill 2 Glencoe Geometry

43 nswers (Lesson 6-1) 6-1 kills ractice roportions Glencoe/cGraw-Hill 27 Glencoe Geometry nswers Lesson FOOLL tight end scored 6 touchdowns in 14 games. Find the ratio of touchdowns per game. 0.43:1 2. UION In a schedule of 6 classes, arta has 2 elective classes. What is the ratio of elective to non-elective classes in arta s schedule? 1:2 3. IOLOG Out of 274 listed species of birds in the United tates, 78 species made the endangered list. Find the ratio of endangered species of birds to listed species in the United tates n artist in ortland, Oregon, makes bronze sculptures of dogs. he ratio of the height of a sculpture to the actual height of the dog is 2:3. If the height of the sculpture is 14 inches, find the height of the dog. 21 in. 5. HOOL he ratio of male students to female students in the drama club at ampbell High chool is 3:4. If the number of male students in the club is 18, what is the number of female students? 24 olve each proportion Find the measures of the sides of each triangle. 12. he ratio of the measures of the sides of a triangle is 3:5:7, and its perimeter is 450 centimeters. 0 cm, 150 cm, 210 cm 13. he ratio of the measures of the sides of a triangle is 5:6:, and its perimeter is 220 meters. 55 m, 66 m, m 14. he ratio of the measures of the sides of a triangle is 4:6:8, and its perimeter is 126 feet. 28 ft, 42 ft, 56 ft 15. he ratio of the measures of the sides of a triangle is 5:7:8, and its perimeter is 40 inches. 10 in., 14 in., 16 in. 6-1 ractice (verage) roportions 1. NUIION One ounce of cheddar cheese contains grams of fat. i of the grams of fat are saturated fats. Find the ratio of saturated fats to total fat in an ounce of cheese. 2:3 2. FING he ratio of goats to sheep at a university research farm is 4:7. he number of sheep at the farm is 28. What is the number of goats? dward Hopper s oil on canvas painting Nighthawks has a length of 60 inches and a width of 30 inches. print of the original has a length of 2.5 inches. What is the width of the print? 1.25 in. olve each proportion Find the measures of the sides of each triangle. 10. he ratio of the measures of the sides of a triangle is 3:4:6, and its perimeter is 104 feet. 24 ft, 32 ft, 48 ft 11. he ratio of the measures of the sides of a triangle is 7::12, and its perimeter is 84 inches. 21 in., 27 in., 36 in. 12. he ratio of the measures of the sides of a triangle is 6:7:, and its perimeter is 77 centimeters. 21 cm, 24.5 cm, 31.5 cm Find the measures of the angles in each triangle. 13. he ratio of the measures of the angles is 4:5:6. 48, 60, he ratio of the measures of the angles is 5:7:8. 45, 63, IG he span of the enjamin Franklin suspension bridge in hiladelphia, ennsylvania, is 1750 feet. model of the bridge has a span of 42 inches. What is the ratio of the span of the model to the span of the actual enjamin Franklin ridge? Glencoe/cGraw-Hill 28 Glencoe Geometry Glencoe/cGraw-Hill 3 Glencoe Geometry

44 nswers (Lesson 6-1) Lesson eading to Learn athematics roportions re-ctivity How do artists use ratios? ead the introduction to Lesson 6-1 at the top of page 282 in your tetbook. stimate the ratio of length to width for the background rectangles in iffany s lematis kylight. ample answer: about 5 to 2 eading the Lesson 1. atch each description in the first column with a word or phrase from the second column. a. he ratio of two corresponding quantities iv i. proportion b. r and u in the equation r s t u v ii. cross products c. a comparison of two quantities vi iii. means d. ru and st in the equation r s t u ii iv. scale factor e. an equation stating that two ratios are equal i v. etremes f. s and t in the equation r s t u iii vi. ratio 2. If m, n, p, and q are nonzero numbers such that m n p, which of the following q statements could be false?,,. np mq. n p m q. mp nq. qm pn. m n q q n p F. p m G. m:p n:q H. m:n p:q 3. Write two proportions that match each description. a. eans are 5 and 8; etremes are 4 and 10. any two of , , , b. eans are 5 and 4; etremes are positive integers that are different from means. any two of or , , , , , , , Helping ou emember 4. ometimes it is easier to remember a mathematical idea if you put it into words without using any mathematical symbols. How can you use this approach to remember the concept of equality of cross products? ample answer: he product of the means equals the product of the etremes. Glencoe/cGraw-Hill 2 Glencoe Geometry 6-1 nrichment Golden ectangles Use a straightedge, compass, and the instructions below to construct a golden rectangle. 1. onstruct square with sides of 2 cm. 2. onstruct the midpoint of. all the midpoint. heck students construction marks. he final figure should look like the one below. 3. raw. et your compass at an opening equal to. Use as the center to draw an arc that intersects. all the point of intersection. 4. onstruct a line through that is perpendicular to. 5. raw so that it intersects the perpendicular line in step 4. all the intersection point Q. Q is a golden rectangle because the ratio of its length to its width is heck this conclusion by finding the value of Q. ectangles whose sides have this ratio are, it is said, the most pleasing to the human eye. Q figure consisting of similar golden rectangles is shown below. Use a compass and the instructions below to draw quarter-circle arcs that form a spiral like that found in the shell of a chambered nautilus. 6. Using as a center, draw an arc that passes through and. 7. Using as a center, draw an arc that passes through and. 8. Using F as a center, draw an arc that passes through and G.. Using H as a center, draw an arc that passes through G and J. 10. Using K as a center, draw an arc that passes through J and L. J L F N H K G 11. Using as a center, draw an arc that passes through L and N. Glencoe/cGraw-Hill 300 Glencoe Geometry Glencoe/cGraw-Hill 4 Glencoe Geometry