NAME DATE PERIOD. Study Guide and Intervention
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1 7-1 tudy Guide and Intervention atios and Proportions Write and Use atios ratio is a comparison of two quantities by divisions. The ratio a to b, where b is not zero, can be written as a b or a:b. ample 1 In 2007 the oston edo baseball team won 6 games out of 162 games played. Write a ratio for the number of games won to the total number of games played. To find the ratio, divide the number of games won by the total number of games played. The result is 6, which is about 0.. The oston edo won about % of their games in hapter esources ample 2 The ratio of the measures of the angles in JHK is 2::4. Find the measures of the angles. The etended ratio 2::4 can be rewritten 2::4. ketch and label the angle measures of the triangle. Then write and solve an equation to find the value of = Triangle um Theorem = ombine like terms. = 20 ivide each side by. The measures of the angles are 2(20) or 40, (20) or 60, and 4(20) or ercises 1. In the 2007 Major eague aseball season, le odriguez hit 4 home runs and was at bat times. What is the ratio of home runs to the number of times he was at bat? 4 or about There are 12 girls in the sophomore class of 0 students. What is the ratio of girls to total students? The length of a rectangle is inches and its width is inches. What is the ratio of length to width? 4. The ratio of the sides of a triangle is :1:17. Its perimeter is 40 inches. Find the length of each side of the triangle. 6 in., in., 204 in.. The ratio of the measures of the three angles of a triangle is 7::20. Find the measure of each angle of the triangle., 4, 0 hapter 7 Glencoe Geometry
2 7-1 tudy Guide and Intervention (continued) atios and Proportions Use Properties of Proportions statement that two ratios are equal is called a proportion. In the proportion a b = c, where b and d are d not zero, the values a and d are 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. This is the ross Product Property. ample 1 16 = 27 olve 16 = 27. = ross Products Property = 42 Multiply. = 4 ivide each side by. a b = c d a d = b c etremes means ample 2 POITI Mayor Hernandez conducted a random survey of 200 voters and found that 1 approve of the job she is doing. If there are 4,000 voters in Mayor Hernandez s town, predict the total number of voters who approve of the job she is doing. Write and solve a proportion that compares the number of registered voters and the number of registered voters who approve of the job the mayor is doing = 4,000 voters who approve all voters 1 4,000 = 200 ross Products Property 6,40,000 = 200 implify. 2,400 = ivide each side by 200. ased on the survey, about 2,400 registered voters approve of the job the mayor is doing. ercises olve each proportion = = y 6 2. = y = If Vs cost $44., find the cost of one V. $14. = 0 = 4-7. OTNY ryon is measuring plants in a field for a science project. Of the first 2 plants he measures, 1 of them are smaller than a foot in height. If there are 4000 plants in the field, predict the total number of plants smaller than a foot in height. 00 plants hapter 7 6 Glencoe Geometry
3 7-1 Practice atios and Proportions 1. NUTITION 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: 2. FMING The ratio of goats to sheep at a university research farm is 4:7. The number of sheep at the farm is 2. What is the number of goats? 16. QUITY ONTO worker at an automobile assembly plant checks new cars for defects. Of the first 20 cars he checks, 4 have defects. If,00 cars will be checked this month, predict the total number of cars that will have defects. 10 cars olve each proportion. 4. = = = = = = The ratio of the measures of the sides of a triangle is :4:6, and its perimeter is 4 feet. Find the measure of each side of the triangle. ft, 2 ft, 4 ft 11. The ratio of the measures of the sides of a triangle is 7::, and its perimeter is 4 inches. Find the measure of each side of the triangle. 21 in., 27 in., 6 in.. The ratio of the measures of the sides of a triangle is 6:7:, and its perimeter is 77 centimeters. Find the measure of each side of the triangle. 21 cm,. cm, 1. cm 1. The ratio of the measures of the three angles is 4::6. Find the measure of each angle of the triangle. 4, 60, The ratio of the measures of the three angles is :7:. Find the measure of each angle of the triangle. 4, 6, IG construction worker is placing rivets in a new bridge. He uses 42 rivets to build the first 2 feet of the bridge. If the bridge is to be 2200 feet in length, predict the number of rivets that will be needed for the entire bridge. 46,200 rivets hapter 7 Glencoe Geometry
4 7-2 tudy Guide and Intervention imilar Polygons Identify imilar Polygons imilar polygons have the same shape but not necessarily the same size. ample 1 If XYZ, list all pairs of congruent angles and write a proportion that relates the corresponding sides. Use the similarity statement. ongruent angles: X, Y, Z Proportion: XY = YZ = ZX ample 2 etermine whether the pair of figures is similar. If so, write the similarity statement and scale factor. plain your reasoning. tep 1 ompare corresponding angles. W P, X Q, Y, Z orresponding angles are congruent. X tep 2 ompare corresponding sides. W Q P WX PQ = = 2, XY Q = 1 = 2, YZ = 1 = 2, and 1 6 Z ercises ZW P = 6 =. ince corresponding sides are proportional, 2 WXYZ PQ. The polygons are similar with a scale factor of 2. ist all pairs of congruent angles, and write a proportion that relates the corresponding sides for each pair of similar polygons. 1. F GHJ G, H, F J; GH = F HJ = F JG 2. PQ TUWX P T, Q U, W, X; PQ TU = Q UW = WX = P XT etermine whether each pair of figures is similar. If so, write the similarity statement and scale factor. If not, eplain your reasoning.. no, 4. hapter 7 Glencoe Geometry PQ PT; scale factor Y
5 7-2 tudy Guide and Intervention (continued) imilar Polygons Use imilar Figures You can use scale factors and proportions to find missing side lengths in similar polygons. ample 1 The two polygons are ample 2 If F GHJ, find the similar. Find and y. scale factor of F to GHJ and the perimeter of each triangle. 2 N The scale factor is 16 1 F T M P HJ = = 2. Use the congruent angles to write the corresponding vertices in order. T MNP Write proportions to find and y = 1 y = = 2(1) 2y = (16) = 26 y = 1 ercises y ach pair of polygons is similar. Find the value of. The perimeter of F is + + or 0. 2 Perimeter of F = Theorem 7.1 Perimeter of GHJ 2 = 0 ubstitution ()(0) = 2 ross Products Property 4 = olve. o, the perimeter of GHJ is 4. esson = 6 = 2.2. If PQ, find the scale factor of to PQ and the perimeter of each polygon. scale factor 4 ; perimeter = 6; perimeter PQ = ; = 1 = hapter 7 1 Glencoe Geometry
6 7-2 Practice imilar Polygons etermine whether each pair of figures is similar. If so, write the similarity statement and scale factor. If not, eplain your reasoning T K P M 2 J Q JKM QP; J Q, K,, M P, and U UVT; U, V, and T by the Third ngle Theorem and JK Q = K = M P = MJ PQ = or 1.67 UV = VT = TU = 2. V ach pair of polygons is similar. Find the value of N M + P. PNTGON If PQT, find the scale factor of to PQT and the perimeter of each polygon. scale factor ; perimeter = 6; 7 perimeter PQT = 1 6. WIMMING POO The Minnitte family and the neighboring Gaudet family both have in-ground swimming pools. The Minnitte family pool, PQ, measures 4 feet by 4 feet. The Gaudet family pool, WXYZ, measures 40 feet by 70 feet. re the two pools similar? If so, write the similarity statement and scale factor. yes; PQ WXYZ; scale factor 6 4. P 4 ft Q 6 F W 4 ft Z ft 21 X 70 ft Y esson 7-2 hapter 7 1 Glencoe Geometry
7 7- tudy Guide and Intervention imilar Triangles Identify imilar Triangles Here are three ways to show that two triangles are similar. imilarity imilarity imilarity Two angles of one triangle are congruent to two angles of another triangle. The measures of the corresponding side lengths of two triangles are proportional. The measures of two side lengths of one triangle are proportional to the measures of two corresponding side lengths of another triangle, and the included angles are congruent. ample 1 etermine whether the ample 2 etermine whether the triangles are similar. triangles are similar. 1 6 F F = 6 = 2 F = = 2 = 1 = 2 F by imilarity. M 70 N 6 Q 4 P 70 4 = 6 MN, so Q = NP. m N = m, so N. NMP Q by imilarity. ercises etermine whether the triangles are similar. If so, write a similarity statement. plain your reasoning. 1.. U T F yes; F; imilarity yes; TU T; imilarity. F 16 N 26 M J 6 K 6. G K W 20 1 no; I P 4 6 J Y yes; KJ PZ; imilarity H Q Z X yes; F NM; imilarity yes; IGH Q; imilarity hapter 7 1 Glencoe Geometry
8 7- tudy Guide and Intervention (continued) imilar Triangles Use imilar Triangles imilar triangles can be used to find measurements. ample 1 F. Find the ample 2 person 6 feet tall casts values of and y. a 1.-foot-long shadow at the same time that a flagpole casts a 7-foot-long shadow. How tall is the flagpole? y F = F 1 = 1 F = F y 1 = 1 1 = (1 ) y = ercises = y = 6 1. ft 6 ft 7 ft? The un s rays form similar triangles. Using for the height of the pole, 6 = 1. 7, so 1. = 42 and = 2. The flagpole is 2 feet tall. G Identify the similar triangles. Then find each measure. 1. J K 2. IU Y X 1 Z J W 1 U XYZ JK; 26 GIU GHW; 0. Q 4. Q 6 2 V 6 20 QV T; ; 1.. M K 6. QP N 7.2 G T V P F 16 M 2 Q P KVM PM; 1 FQ FNP; 2 7. The heights of two vertical posts are 2 meters and 0.4 meter. When the shorter post casts a shadow that is 0. meter long, what is the length of the longer post s shadow to the nearest hundredth?.7 m 0 I H esson 7- hapter 7 1 Glencoe Geometry
9 7- Practice imilar Triangles etermine whether the triangles are similar. If so, write a similarity statement. If not, what would be sufficient to prove the triangles similar? plain your reasoning. 1. J Y 1 42 K W 42 yes; JK WY; imilarity No. The triangles would be similar by or if MN PQ. G Identify the similar triangles. Then find each measure.. M, QP 4. N, M + M 1 N Q - 1 P J N + M K MN PQN; M = ; QP = JN KM; N = 21; M = 14. P, P 6. G, HG TP QP; P = ; P = 16 GF HGI; G = 6; HG = 7. INIT MUMNT lighthouse casts a -foot shadow. nearby lamppost that measures feet inches casts an -foot shadow. a. Write a proportion that can be used to determine the height of the lighthouse. ample answer: If = height of lighthouse,.2 =. b. What is the height of the lighthouse? 4 ft + 4 esson 7- hapter 7 21 Glencoe Geometry
10 7-4 tudy Guide and Intervention Parallel ines and Proportional Parts Proportional Parts within Triangles In any triangle, a line parallel to one side of a triangle separates the other two sides proportionally. This is the Triangle Proportionality Theorem. The converse is also true. If XY, then X XT = Y YT. If X XT = Y, then XY YT. If X and Y are the midpoints of T and T, then XY is a midsegment of the triangle. The Triangle Midsegment Theorem states that a midsegment is parallel to the third side and is half its length. If XY is a midsegment, then XY and XY = 1 2. X T Y ample 1 In, F. Find. ample In GHJ, HK =, KG =, and J is one-half the length of G. Is HK K? + 22 F + 2 ince F, F F = = = = = ercises G Find the value of Using the converse of the Triangle Proportionality Theorem, show that HK KG = J G. et J = and G = 2. HK KG = = 1 2 J G = 2 = 1 2 ince 1 2 = 1, the sides are proportional and 2 HJ K hapter 7 Glencoe Geometry
11 7-4 tudy Guide and Intervention (continued) Parallel ines and Proportional Parts Proportional Parts with Parallel ines 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 l 1 c d l 2 l u v n w m l 4 l l 6 If l 1 l 2 l, then a b = c d. If l 4 l l 6 and u v = 1, then w = 1. ample efer to lines l 1, l 2, and l above. If a =, b =, and c =, find d. l 1 l 2 l so = d. Then d = 40 and d = 1 1. ercises G Find and y = = y - 1 2y + 2 = ; y = y y + 2 y = 1 = y = 16 y 2 y + esson 7-4 hapter 7 2 Glencoe Geometry
12 If JK = 7, KH = 21, and J = 6, 2. If U =, U = 14, TV = - 1, find I. and V = 17., find and TV. H kills Practice Parallel ines and Proportional Parts K J U V I 1 11; etermine whether. Justify your answer. T. = 1, =, =, and = yes; = = 4 4. =, = 27, and = 1 no;. = 0, = 4, and = 2 yes; = = 2 1 JH is a midsegment of KM. Find the value of G Find and y y y y - 1. y - = 6, y = 6. = 2, y = 4 hapter 7 26 Glencoe Geometry
13 7- tudy Guide and Intervention Parts of imilar Triangles pecial egments of imilar Triangles When two triangles are similar, corresponding altitudes, angle bisectors, and medians are proportional to the corresponding sides. ample In the figure, XYZ, with angle bisectors as shown. Find. ince XYZ, the measures of the angle bisectors are proportional to the measures of a pair of corresponding sides. XY = YW = = () = 12 = 1.2 X Y W Z ercises Find hapter 7 0 Glencoe Geometry
14 7- tudy Guide and Intervention (continued) Parts of imilar Triangles Triangle ngle isector Theorem n angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides. ample Find. ince U is an angle bisector, U TU = T. 20 = = 20(1) 0 = 00 = 1 0 U 20 T ercises Find the value of each variable n 6 42 r a esson 7- hapter 7 1 Glencoe Geometry
15 7- kills Practice Parts of imilar Triangles Find If T FG, H is an 6. If MNP, is an altitude of T, FJ is an altitude of altitude of, MQ is an altitude of FG, T = 6, H =, and FJ = 7, MNP, =, = 14, and find FG. MQ =., find MN. F M H T.4 1 Find the value of each variable. 7.. m 26 J G 20 Q P N hapter 7 2 Glencoe Geometry
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