Geometry. Chapter 6 Resource Masters

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1 Geometry hapter 6 esource Masters

6-4 NM PIO tudy Guide and Intervention Parallel Lines and Proportional Parts Proportional Parts of riangles In any triangle, a line parallel to one side of a triangle separates the other two sides

2 6-4 NM PIO tudy Guide and Intervention Parallel Lines and Proportional Parts Proportional Parts 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 Y are the midpoints of and, then X Y is a midsegment of the triangle. he riangle Midsegment heorem states that a midsegment is parallel to the third side and is half its length. If XY X Y,then X Y. If X Y,then XY. X Y If X Y is a midsegment, then XY 1 and XY 2. X Y ample 1 ample F 2 F ince F, F In, F. Find. triangle has vertices (3, 6), ( 3, 2), and F(7, 2). Midsegment G H is parallel to F. Find the length of G H. G H is a midsegment, so its length is onehalf that of F. Points and F have the same y-coordinate, so F 7 ( 3) 10. he length of midsegment G H is 5. ercises Find Lesson In ample 2, find the slope of F and show that F G H. Glencoe/McGraw-Hill 313 Glencoe Geometry

6-4 NM PIO tudy Guide and Intervention (continued) Parallel Lines and Proportional Parts ivide egments Proportionally When three or more parallel lines cut two

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 4 5 6 If 1 2 3, then a b c.

3 6-4 NM PIO tudy Guide and Intervention (continued) Parallel Lines and Proportional Parts ivide egments Proportionally 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/McGraw-Hill 314 Glencoe Geometry

4 NM PIO 6-4 kills Practice Parallel Lines and Proportional Parts 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 , 27, and is one third of 5. 30, 45, and is twice OOIN GOMY For ercises 6 8, use the following information. riangle has vertices ( 5, 2), (1, 8), and (4, 2). Point 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 Find and y. 10. Find and y y 8 7 y y 1 3y 5 Glencoe/McGraw-Hill 315 Glencoe Geometry

5 If 24, 27, and 18, 2. Find, Q, and if Q 6, find. 12, P 27, and 4. NM PIO Practice Parallel Lines and Proportional Parts P Q etermine whether J K N M. 3. JN 18, JL 30, KM 21, and ML 35 J K N M L 4. KM 24, KL 44, and NL 5 6 JN OOIN GOMY For ercises 5 and 6, use the following information. riangle FG has vertices ( 4, 1), F(2, 5), and G(2, 1). Point K is the midpoint of G and H is the midpoint of F G. 5. how that F is parallel to K H. 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 MP he distance from Wilmington to sh Grove along Kendall is 820 feet and along Magnolia, 660 feet. If the distance between eech and sh Grove along Magnolia is 280 feet, what is the distance between the two streets along Kendall? Magnolia Wilmington eech sh Grove Kendall Glencoe/McGraw-Hill 316 Glencoe Geometry

6 6-4 NM PIO eading to Learn Mathematics Parallel Lines and Proportional Parts Pre-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 Michigan venue than along Lake hore rive. eading the Lesson 1. Provide 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 You 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/McGraw-Hill 317 Glencoe Geometry

7 6-4 NM PIO nrichment Parallel Lines and ongruent Parts 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. Mark where the lines of the notebook paper meet the segment. Label the fifth point, P. P tep 3 raw P. hrough each of the other marks on P, construct a line parallel to P. he points where these lines intersect will divide into five congruent segments. P 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/McGraw-Hill 318 Glencoe Geometry

8 6-5 NM PIO tudy Guide and Intervention Parts of imilar riangles Perimeters 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. XYZ 2. Y X Z XYZ N Z M 20 8 P X Y P M 10 Lesson 6-5 Glencoe/McGraw-Hill 319 Glencoe Geometry

9 6-5 NM PIO tudy Guide and Intervention (continued) Parts 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, XYZ, with angle bisectors as shown. Find. ample 1 ample ince XYZ, the measures of the angle bisectors are proportional to the measures of a pair of corresponding sides. XY Y W (8) ercises X Y 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/McGraw-Hill 320 Glencoe Geometry

10 6-5 NM PIO kills Practice Parts 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. PQ, if PQ LMN, LM 16, 4. KLM, if KLM FGH, FG 30, MN 14, NL 27, and P 18 GH 38, HF 38, and KL 24 L M P N Q H M G L K F Use the given information to find each measure. 5. Find FG if FG, H is an 6. Find MN if MNP, is an altitude of, F J is an altitude of altitude of, M Q is an altitude of FG, 6, H 5, and FJ 7. MNP, 24, 14, and MQ F M H J G Q P N Find. 7. HKL XYZ 8. K Y 10 H 18 L X 15 Z Lesson 6-5 Glencoe/McGraw-Hill 321 Glencoe Geometry

11 6-5 NM PIO Practice Parts of imilar riangles Find the perimeter of the given triangle. 1. F, if F, 36, 2. U, if U KLM, KL 12, 20, 40, and 35 LM 31, MK 32, and U 28 M U L F K Use the given information to find each measure. 3. Find P if JKL NP, K M is an 4. Find ZY if U XYZ, U is an altitude of JKL, P is an altitude of altitude of U, Z is an altitude of NP, KL 28, KM 18, and XYZ, U 8.5, U 6, and P Z K P Z U J M L N X Y Find PHOOGPHY 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/McGraw-Hill 322 Glencoe Geometry

12 6-5 NM PIO eading to Learn Mathematics Parts of imilar riangles Pre-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. P M X U Q W N Y V i. efinition of congruent polygons ii. efinition of similar polygons iii. Proportional Perimeters 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. M UN? VW b. U W?X d. UV? e. P? P f. U N UW? P g. W Q? h. U W? Q VW V Helping You 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/McGraw-Hill 323 Glencoe Geometry

13 6-5 NM PIO nrichment Proportions 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. You can write many proportions by identifying similar triangles in the following diagram. In the diagram, M N, FL M, and Q. F G H J K L M N P Q nswer each question. Use the diagram above. 1. Name a triangle similar to GNP. 2. Name a triangle similar to JH. 3. Name two triangles similar to JK. 4. Name a triangle similar to P. omplete each proportion. 5. G F P? 8. P H P P G? P 6. H?? 9. J L J? 7. J J L 10. M N? MP P 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 MN 12, NP 32, and P 48, find G. ound to the nearest tenth. 14. If H 18, HP 82, and 130, find J. 15. Write three more problems that can be solved using the diagram above. Glencoe/McGraw-Hill 324 Glencoe Geometry

14 NM PIO 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/McGraw-Hill 325 Glencoe Geometry

15 6-6 Nongeometric Iteration n iterative process can be applied to an algebraic epression or equation. he result is called a recursive formula. ample NM PIO 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/McGraw-Hill 326 Glencoe Geometry

16 6-6 NM PIO kills Practice 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/McGraw-Hill 327 Glencoe Geometry

17 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 NM PIO Practice Fractals and elf-imilarity MH tage 2 MH MH 1. raw tages 3 and 4 of the book cover fractal. tage 3 tage 4 MH MH MH MH MH MH MH 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 $96,000. 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/McGraw-Hill 328 Glencoe Geometry

18 NM PIO 6-6 eading to Learn Mathematics Fractals and elf-imilarity Pre-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. Match 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.) Phrase 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 You 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/McGraw-Hill 329 Glencoe Geometry

19 6-6 NM PIO nrichment he Möbius trip Möbius strip is a special surface with only one side. It was discovered by ugust Ferdinand Möbius, a German astronomer and mathematician. 1. o make a Möbius strip, cut a strip of paper about 16 inches long and 1 inch wide. Mark 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 Möbius strip down the middle? ry it. 4. Make another Möbius 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/McGraw-Hill 330 Glencoe Geometry

NAME DATE PERIOD. 10. The ratio of the measures of the sides of a triangle is 3:4:6, and its perimeter is 104 feet.

NAME DATE PERIOD. 10. The ratio of the measures of the sides of a triangle is 3:4:6, and its perimeter is 104 feet. 6-1 IO ractice roportions 1. IIO One ounce of cheddar cheese contains grams of fat. i of the grams of fat are saturated fats. ind the ratio of saturated fats to total fat in an ounce of cheese.. I he ratio