North Valleys High School Mathematics Geometry 2

Transcription

1 North Valleys High School Mathematics Geometry 2 Student Name: Period: The Topics for this section: Solving Proportions based on Triangles ilations Similarity and Similar Triangles Triangles ivided by Parallel Lines Midsegments of Triangles Proving the Pythagorean Theorem t the end of this section the student should: Solve proportions arithmetically and algebraically pply proportions to a variety of triangle solutions Understand the rules for triangle proportions and midsegments Know and apply the rules for similarity raw dilations and do calculations based on dilations Understand the basis of the Pythagorean Theorem from a ompleted Proof

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3 ontents Title Page Solving Proportions 1 Practice 7-1 Ratios and Proportions 2 ilation Practice 1 3 ilation Practice 2 4 Skills Practice 7-2 Similar Polygons 6 Skills Practice/Practice 7-3 Similar Triangles 8 Word Problems 7-3 Similar Triangles Similarity 2 11 Skills Practice/Practice 7-5 Parts of Similar Triangles 13 Word Problems 7-5 Parts of Similar Triangles 15 Solving Triangles with Proportional Parts 16 Triangles with Parallel Lines (Side Splitter) 1 19 Triangles with Parallel Lines (Side Splitter) 2 20 Midsegments 1 21 Midsegments 2 23

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5 g M2T01v2X 3KuustNa osmoqfxtmwzagrmeg LbLq.3 K ijlul kriirghtosx UrEe9srefrhvneQdZ.G h gmgasdheu 1wi6tph mignsftilnzibtqe0 XGHe0otme1tlrXym.z Worksheet by Kuta Software LL North Valleys High School Geometry 1 Solving Proportions W s2w0d172 dkyuqtaag hs6ouflt1waasrqeo klsll4.m 7 t4lil2 IrNi9gGhctosg sr5epsvetr8vteydq.v Solve each proportion. I: 1 Name ate Period 1) v 2) r 3) a 4) n 5) 6) 7) n 8) k 9) ) p 11) m m 12) n n 13) r 14) r 15) 16) v v 17) b b 18) n n 19) 20) n n Page 1 of 24

6 NME TE PERIO 7-1 Practice Ratios and Proportions 1. NUTRITION One ounce of cheddar cheese contains 9 grams of fat. Si of the grams of fat are saturated fats. Find the ratio of saturated fats to total fat in an ounce of cheese. 2. FRMING The ratio of goats to sheep at a university research farm is 4:7. The number of sheep at the farm is 28. What is the number of goats? 3. QULITY ONTROL worker at an automobile assembly plant checks new cars for defects. Of the first 280 cars he checks, 4 have defects. If,500 cars will be checked this month, predict the total number of cars that will have defects. Solve each proportion = = = = = = The ratio of the measures of the sides of a triangle is 3:4:6, and its perimeter is 4 feet. Find the measure of each side of the triangle. 11. The ratio of the measures of the sides of a triangle is 7:9:12, and its perimeter is 84 inches. Find the measure of each side of the triangle. 12. The ratio of the measures of the sides of a triangle is 6:7:9, and its perimeter is 77 centimeters. Find the measure of each side of the triangle. 13. The ratio of the measures of the three angles is 4:5:6. Find the measure of each angle of the triangle. 14. The ratio of the measures of the three angles is 5:7:8. Find the measure of each angle of the triangle. 15. RIGES 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. opyright Glencoe/McGraw-Hill, a division of The McGraw-Hill ompanies, Inc. hapter 7 8 Glencoe Geometry Page 2 of 24

7 ilations () Instructions: raw and label the dilated image for each triangle. Label the center of dilation. 1) 2) 3) 4) 5) 6) MTH-RILLS.OM MTH-RILLS.OM MTH-RILLS.OM MTH-RILLS.OM Page 3 of 24

8 Geometry Practice: G.G.58 #1 P.I. G.G.58: efine, investigate, justify, and apply similarities (dilations and the composition of dilations and isometries) NME: 4. Graph with (4, 3) and (1, 1). Then graph its dilation with a scale factor of Graph EF with E( 2, 2) and F(3, 4). Then graph its dilation with a scale factor of 2. y y [1] 2. Graph GH with G(3, 3) and H( 2, 1). Then graph its dilation with a scale factor of 2. [4] 5. Graph JK with J(2, 3) and K( 4, 4). Then graph its dilation with a scale factor of 2. y y [2] 3. Graph IJ with I( 4, 4) and J(1, 3). Then graph its dilation with a scale factor of 2. [5] 6. Graph GH with G(4, 4) and H( 2, 3). Then graph its dilation with a scale factor of 2. y y [6] [3] Page 4 of 24

9 Geometry Practice: G.G.58 # Graph with (3, 2) and ( 1, 3). Then graph its dilation with a scale factor of 1.5. NME:. Graph E with (1, 1) and E(2, 2). Then graph its dilation with a scale factor of 1.5. y y [7] 8. Graph with ( 4, 2) and (1, 4). Then graph its dilation with a scale factor of 2.5. [] 11. Graph with (4, 1) and (3, 3). Then graph its dilation with a scale factor of 2.5. y y [8] 9. Graph FG with F(3, 4) and G(2, 1). Then graph its dilation with a scale factor of 0.5. [11] 12. Graph EF with E( 2, 4) and F( 1, 2). Then graph its dilation with a scale factor of 0.5. y y [9] [12] Page 5 of 24

10 NME TE PERIO 7-2 etermine whether each pair of figures is similar. If so, write the similarity statement and scale factor. If not, eplain your reasoning. E W 7.5 X P 3 Q Skills Practice Similar Polygons F 3 S 3 3 R Z Y 7.5 Each pair of polygons is similar. Find the value of E 7 H F S M + 1 T W 8 P 9 L 5 U V R 3 Q 13 G Y 4 W 3 4 X N P S T 9 U T - 1 S opyright Glencoe/McGraw-Hill, a division of The McGraw-Hill ompanies, Inc. hapter 7 14 Glencoe Geometry Page 6 of 24

11 NME TE PERIO 7-2 Practice Similar Polygons etermine whether each pair of figures is similar. If so, write the similarity statement and scale factor. If not, eplain your reasoning T L K P S M 25 J Q 12 R 12 U 24 V Each pair of polygons is similar. Find the value of N M + 9 P L F + 1 E 40-3 Lesson 7-2 opyright Glencoe/McGraw-Hill, a division of The McGraw-Hill ompanies, Inc. 5. PENTGONS If E PQRST, find the scale factor of E to PQRST and the perimeter of each polygon. 6. SWIMMING POOLS The Minnitte family and the neighboring Gaudet family both have in-ground swimming pools. The Minnitte family pool, PQRS, measures 48 feet by 84 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. P 48 ft Q ft 20 S R W Z 40 ft 21 X 70 ft Y hapter 7 15 Glencoe Geometry Page 7 of 24

12 NME TE PERIO 7-3 etermine whether each pair of triangles is similar. If so, write a similarity statement. If not, what would be sufficient to prove the triangles similar? Eplain your reasoning R Q 12 Skills Practice Similar Triangles P T U S P J 21 M S 60 K M Q 30 T R LGER Identify the similar triangles. Then find each measure JL J E K EH 8. VT 9 F H S 3-3 E V G 9 R T U L - 3 N 4 M opyright Glencoe/McGraw-Hill, a division of The McGraw-Hill ompanies, Inc. hapter 7 20 Glencoe Geometry Page 8 of 24

13 NME TE PERIO 7-3 Practice Similar Triangles etermine whether the triangles are similar. If so, write a similarity statement. If not, what would be sufficient to prove the triangles similar? Eplain your reasoning. 1. J Y S K W LGER Identify the similar triangles. Then find each measure. 3. LM, QP 4. NL, ML L + 3 M 18 N 12 Q - 1 P J N + 5 M 8 K L opyright Glencoe/McGraw-Hill, a division of The McGraw-Hill ompanies, Inc INIRET MESUREMENT lighthouse casts a 128-foot shadow. nearby lamppost that measures 5 feet 3 inches casts an 8-foot shadow. a. Write a proportion that can be used to determine the height of the lighthouse. b. What is the height of the lighthouse? Lesson 7-3 hapter 7 21 Glencoe Geometry Page 9 of 24

14 NME TE PERIO 7-3 Word Problem Practice Similar Triangles 1. HIRS local furniture store sells two versions of the same chair: one for adults, and one for children. Find the value of such that the chairs are similar. 18 X 4. SHOWS radio tower casts a shadow 8 feet long at the same time that a vertical yardstick casts a shadow half an inch long. How tall is the radio tower? OTING The two sailboats shown are participating in a regatta. Find the value of in. 220 in MOUNTIN PEKS Gavin and rianna want to know how far a mountain peak is from their houses. They measure the angles between the line of site to the peak and to each other s houses and carefully make the drawing shown. Gavin in. 83 rianna 2 in in. Peak X in. 264 in. 3. GEOMETRY Georgia draws a regular pentagon and starts connecting its vertices to make a 5-pointed star. fter drawing three of the lines in the star, she becomes curious about two triangles that appear in the figure, and E. They look similar to her. Prove that this is the case. E The actual distance between Gavin and rianna s houses is miles. a. What is the actual distance of the mountain peak from Gavin s house? Round your answer to the nearest tenth of a mile. b. What is the actual distance of the mountain peak from rianna s house? Round your answer to the nearest tenth of a mile. opyright Glencoe/McGraw-Hill, a division of The McGraw-Hill ompanies, Inc. hapter 7 22 Glencoe Geometry Page of 24

15 Geometry Practice: G.G.45 #2 NME: P.I. G.G.45: Investigate, justify, and apply theorems about similar triangles 1. omplete the following when IJK ~ LMN. a) m K = m b) KI JI = NL? 2. In QRS, QR7, RS 13, and mr46. In UVT, VT14, TU 28, and mt46. State whether the triangles are similar, and if so, write a similarity statement. 3. Find the scale factor that maps onto if ( 2, 0), ( 0, 4), ( 6, 0), ( 30, ), ( 0, 6), ( 9, 0 ). How are the figures related? Eplain. 4. with vertices ( 21, ), ( 25, ), and ( 2, 4 ) is similar to MNO with vertices M( 3, 1) and N (, 3 9). Find four possibilities for the coordinates of verte O. 5. lamppost is 6 feet high and casts an 8-foot shadow. t the same time of day, a flagpole directly behind the lamppost casts a 28-foot shadow. H 6 ft 28 ft 8 ft Which proportion can be used to find the height, H, of the flagpole? [] H 28 6 = [] 8 H 8 6 = [] 28 8 = H [] 28 6 H 28 = 8 6 Page 11 of 24

16 Geometry Practice: G.G.45 #2 NME: 6. t the same time of day, a man who is 52.8 inches tall casts a 68.8-inch shadow and his son casts a 43-inch shadow. What is the height of the man s son? 52.8 in in. 43 in. [] 33 in. [] 85.8 in. [] in. [] 34 in. 7. Two ladders are leaning against a wall at the same angle as shown. How far up the wall does the shorter ladder reach? 48 ft 30 ft 16 ft [] 8 ft [] ft [] 6 ft [] 20 ft Page 12 of 24

17 NME TE PERIO 7-5 Skills Practice Parts of Similar Triangles Find If RST EFG, SH is an 6. If MNP, is an altitude of RST, FJ is an altitude of altitude of, MQ is an altitude of EFG, ST = 6, SH = 5, and FJ = 7, MNP, = 24, = 14, and find FG. MQ =.5, find MN. F S M R H T E J G Q P N Find the value of each variable m opyright Glencoe/McGraw-Hill, a division of The McGraw-Hill ompanies, Inc hapter 7 32 Glencoe Geometry Page 13 of 24

18 NME TE PERIO 7-5 Practice LGER Find. Parts of Similar Triangles opyright Glencoe/McGraw-Hill, a division of The McGraw-Hill ompanies, Inc. 5. If JKL NPR, KM is an altitude of JKL, PT is an altitude of NPR, KL = 28, KM = 18, and PT = 15.75, find PR. J K M L N P T R 6. If STU XYZ, U is an altitude of STU, Z is an altitude of XYZ, UT = 8.5, U = 6, and Z = 11.4, find ZY. 7. PHOTOGRPHY Francine has a camera in which the distance from the lens to the film is 24 millimeters. a. 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.) b. Suppose 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? S T U X Y Z Lesson 7-5 hapter 7 33 Glencoe Geometry Page 14 of 24

19 NME TE PERIO 7-5 Word Problem Practice Parts of Similar Triangles 1. FLGS n oceanliner is flying two similar triangular flags on a flag pole. The altitude of the larger flag is three times the altitude of the smaller flag. If the measure of a leg on the larger flag is 45 inches, find the measure of the corresponding leg on the smaller flag. 2. TENTS Jana went camping and stayed in a tent shaped like a triangle. In a photo of the tent, the base of the tent is 6 inches and the altitude is 5 inches. The actual base was 12 feet long. What was the height of the actual tent? 4. FLG POLES flag pole attached to the side of a building is supported with a network of strings as shown in the figure. F E The rigging is done so that E = EF, =, and =. What is the ratio of F to E? 5. OPIES Gordon made a photocopy of a page from his geometry book to enlarge one of the figures. The actual figure that he copied is shown below. 3. PLYGROUN The playground at Hank s school has a large right triangle painted in the ground. Hank starts at the right angle corner and walks toward the opposite side along an angle bisector and stops when he gets to the hypotenuse. 30 ft ft 40 ft How much farther from Hank is point versus point? Median 30 mm 39 mm ltitude 29 mm The photocopy came out poorly. Gordon could not read the numbers on the photocopy, although the triangle itself was clear. Gordon measured the base of the enlarged triangle and found it to be 200 millimeters. a. What is the length of the drawn altitude of the enlarged triangle? Round your answer to the nearest millimeter. b. What is the length of the drawn median of the enlarged triangle? Round your answer to the nearest millimeter. opyright Glencoe/McGraw-Hill, a division of The McGraw-Hill ompanies, Inc. hapter 7 34 Glencoe Geometry Page 15 of 24

20 North Valleys High School Geometry 1 I: 1 Name Solving Triangles with Proportional Parts R g2v0n1t21 rktutcat csnoqfvtxwtazrlee 9LmL5.S t ivliln QrJi6gWhftzsL rwe7s2errtvte0dr.h Solve for. ate Period 1) 2) 3) 4) 5) 6) 7) 8) Page 16 of u g2y0s1d2l 9Kgu3t1ai QSroefrtewUaZr4e9 ulmla.r v E8lRlM dreiugkhmtgs5 r6egsve5ryvuendl.m I mmqa1dkeq Yw1iVtuh eitnnfzi4nniet3et og4ewo7m3ertgrym.1 Worksheet by Kuta Software LL

21 -2-3 k2201u2f 9K4uwtKa S4oKft1wva4rZeT olslze. v TUldlY JrjiTgGhatls6 vrfesseegrovfevdw.w 6 vmuaudyei 4wiWt5hY XIunafiJnTirtkej 4GNePo9mneNt8ryc.u Worksheet by Kuta Software LL 9) ) 11) 12) 13) 14) 15) 16) Page 17 of 24

22 -3- Q g2p0t182f kklutt6at XSSoMfMtuwua5rrej yl8lnf.c U Sml5li rr6ihg2htoss roeessegrnvjeldh.8 M MYaodeef 4wiytuh tixn2ftihnvihtheu KGeyoUm5eitQrSy3.Z Worksheet by Kuta Software LL 17) 18) 19) 20) 21) 22) 23) 24) Page 18 of 24

23 Regents Eam Questions G.G.46: Side Splitter Theorem Name: G.G.46: Side Splitter Theorem: Investigate proportions among segments of sides of the triangle, given line(s) parallel to one side and intersecting the other sides of the triangle 1 In the accompanying diagram of equilateral triangle, E 5 and E. 3 In the diagram below of, E is a point on and is a point on, such that E. If E 3, E 6, and 15, find the length of E. If is three times as long as E, what is the perimeter of quadrilateral E? 1) 20 2) 30 3) 35 4) 40 2 In, point is on, and point E is on such that E. If 2, 7, and E 3, what is the length of? 1) 8 2) 9 3).5 4) In the diagram below of E, is a point on E and is a point on such that E, 3, E 20, 16, and 2 2. Find the length of. 1 Page 19 of 24

24 Geometry Practice: G.G.46 P.I. G.G.46: Investigate, justify, and apply theorems about proportional relationships among the segments of the sides of the triangle, given one or more lines parallel to one side of a triangle and intersecting the other two sides of the triangle 1. Given: PQ. Find the length of. NME: 4. Given: PQ. Find the measure of Q P 9 Q 4 6 P Q Given E, solve for. [] 14 [] 11 [] 16 [] In the figure shown, E, = 2 yards, = 9 yards, E = 36 yards, and E = 36 yards. Find. E Given E, solve for. E 8 [] 9 yd [] 8 yd [] 6 yd [] 27 yd E In the figure shown, E, = 7 yards, = 8 yards, E = 4 yards, and E = 16 yards. Find E. 7. Find E Page 20 of 24

25 Regents Eam Questions G.G.42: Midsegments Name: G.G.42: Midsegments: Investigate, justify, and apply theorems about geometric relationships, based on properties of the segment joining the midpoints of two sides of the triangle 1 If the midpoints of the sides of a triangle are connected, the area of the triangle formed is what part of the area of the original triangle? 1 1) 4 1 2) 3 3 3) 8 1 4) 2 3 In the diagram below of, E is a midsegment of, E 7,, and 13. Find the perimeter of. 2 In the diagram below of, is the midpoint of, and E is the midpoint of. 4 In the diagram of below,, 14, and 16. Find the perimeter of the triangle formed by connecting the midpoints of the sides of. If 4, which epression represents E? 1) 2.5 2) 2 5 3) 2 4) Page 21 of 24

26 Regents Eam Questions G.G.42: Midsegments Name: 5 In the diagram below of T, is the midpoint of, O is the midpoint of T, and G is the midpoint of T. If, T 18, and T 22, what is the perimeter of parallelogram OG? 1) 21 2) 25 3) 32 4) 40 6 On the set of aes below, graph and label EF with vertices at (4,4), E(2,2), and F(8,2). If G is the midpoint of EF and H is the midpoint of F, state the coordinates of G and H and label each point on your graph. Eplain why GH E. 2 Page 22 of 24

27 Geometry Practice: G.G.42 P.I. G.G.42: Investigate, justify, and apply theorems about geometric relationships, based on the properties of the line segment joining the midpoints of two sides of the triangle 1. Solve for given = and E = NME: 3. Solve for given = 5 2 and E = 9 6. ssume is the midpoint of and is the midpoint of E ssume is the midpoint of and is the midpoint of E. E 4. Solve for given = 4 2 and E = E 6 8. ssume is the midpoint of and is the midpoint of E. [] 2 [] 2 9 [] 1 2 [] Solve for given = and E = 3 6. ssume is the midpoint of and is the midpoint of E. E E [] 7 2 [] 1 2 [] 2 7 [] 2 Page 23 of 24

28 Geometry Practice: G.G Find the area of the rectangle if = 11 and = 22. NME: 8. Find the area of the rectangle if = 16 and = 24. [] 121 [] 33 [] 242 [] Find the area of the rectangle if = 15 and = The vertices of a triangle are ( 3, 2), (3, 4), and (1, 6). Find the coordinates of S, the midpoint of, and T, the midpoint of. Verify that ST 1 and that ST 2.. Find the values of and y. 4 7 [] 39 [] 360 [] 180 [] 90 y Find the area of the rectangle if = 14 and = 30. [] 3 1, 2 y 25 [] 3 1, 2 y [] 3 1, 2 y 24 [] none of the above Page 24 of 24