Lesson 9.1 The Theorem of Pythagoras


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1 Lesson 9.1 The Theorem of Pythagoras Give all answers rounded to the nearest 0.1 unit. 1. a. p. a 75 cm 14 cm p 6 7 cm 8 cm 1 cm rea 9 in 5. Find the area. 6. Find the coordinates of h and the radius of circle. y 6 in. h 6 7 (7, 1) (11, 4) 7. Find the area. 8. RS cm. Find RV. 7.7 cm R 6.8 cm S T U V 1.4 cm 9. ase area 16 cm and slant height 10. Given PQR, with m P 90, PQ 0 in., cm. What s wrong with this picture? and PR 15 in., find the area of PQR, the length of the hypotenuse, and the altitude to the hypotenuse. Discovering Geometry Practice Your Skills HPTER Kendall Hunt Publishing
2 Lesson 9. The onverse of the Pythagorean Theorem ll measurements are in centimeters. Give answers rounded to the nearest 0.01 cm. In Eercises 1 4, determine whether a triangle with the given side lengths is a right triangle , 10, 98. 1, 04, , 1.4, , 8, Find the area of. 6. What s wrong with this picture? Find. Eplain your method. 8. Find the area of D D D 6 8 In Eercises 9 11, determine whether D is a rectangle and justify your answer. If not enough information is given, write cannot be determined. 9., 4, and 6. D 10., 4, D 4, and , 4, D, D 4, and D. 60 HPTER 9 Discovering Geometry Practice Your Skills 008 Kendall Hunt Publishing
3 Lesson 9. Two Special Right Triangles Give your answers in eact form unless otherwise indicated. ll measurements are in centimeters. In Eercises 1, find the unknown lengths. 1. a. a, b. a, b a 14 1 b 0 a b 6 a Find the area of rectangle 5. Find the perimeter and 6.,, D. area of KLMN. and area K 45 N 7 M 1 0 L D 7. Find the area of an isosceles trapezoid if the bases have lengths 1 cm and 18 cm and the base angles have measure 60. In Eercises 8 and 9, find the coordinates of. 8. y 9. y 10 (1, 0) 10 (1, 0) (0, 1) (0, 1) 10. Sketch and label a figure to demonstrate that 18 is equivalent to. Discovering Geometry Practice Your Skills HPTER Kendall Hunt Publishing
4 Lesson 9.4 Story Problems 1. 0 ladder reaches a window 18 high. How far is the foot of the ladder from the base of the building? How far must the foot of the ladder be moved to lower the top of the ladder by?. Robin and Dovey have four pet pigeons that they train to race. They release the birds at Robin s house and then drive to Dovey s to collect them. To drive from Robin s to Dovey s, because of oneway streets, they go.1 km north, turn right and go 1.7 km east, turn le and go. km north, turn right and go 0.9 km east, turn le and go 1. km north, turn le and go 4.1 km west, and finally turn le and go 0.4 km south. How far do the pigeons have to fly to go directly from Robin s house to Dovey s house?. Hans needs to paint the 18 in.wide trim around the roof eaves and gable ends of his house with coats of paint. quart can of paint covers 175 and costs $9.75. gallon can of paint costs $7.95. How much paint should Hans buy? Eplain. 18 in What are the dimensions of the largest triangle that will fit inside a triangle with leg length 14 in.? Sketch your solution. 6 HPTER 9 Discovering Geometry Practice Your Skills 008 Kendall Hunt Publishing
5 Lesson 9.5 Distance in oordinate Geometry In Eercises 1, find the distance between each pair of points. 1. ( 5, 5), (1, ). ( 11, 5), (5, 7). (8, ), ( 7, 6) In Eercises 4 and 5, use the distance formula and the slope of segments to identify the type of quadrilateral. Eplain your reasoning. 4. (, 1), (, ), (8, 1), D(, 4) 5. T(, ), U(4, 4), V(0, 6), W( 5, 1) For Eercises 6 and 7, use with coordinates (4, 14), (10, 6), and (16, 14). 6. Determine whether is scalene, isosceles, or equilateral. Find the perimeter of the triangle. 7. Find the midpoints M and N of and, respectively. Find the slopes and lengths of MN and. How do the slopes compare? How do the lengths compare? 8. Find the equation of the circle with center ( 1, 5) and radius. 9. Find the center and radius of the circle whose equation is (y ) P is the center of the circle. What s wrong with this picture? y (4, 6) P(10, 1) (5, 5) (16, ) Discovering Geometry Practice Your Skills HPTER Kendall Hunt Publishing
6 Lesson 9.6 ircles and the Pythagorean Theorem In Eercises 1 and, find the area of the shaded region in each figure. ll measurements are in centimeters. Write your answers in terms of and rounded to the nearest 0.1 cm. 1. O Tangent PT, QM 1, m P 0 T O P S M Q. P 6 cm. Radius of circle O 7 cm. How far is from the circumference of the circle? O P 4. Two perpendicular chords with lengths 1. cm and 8.8 cm have a common endpoint. What is the area of the circle? 5. D is inscribed in a circle. is a diameter. If 9.6 cm, 5.7 cm, and D.1 cm, find D. 6. Find ST. 7. The coordinate of point M is, 1. Find the measure of OM. y O 0 T 6 R (0, 1) P S M O (1, 0) 64 HPTER 9 Discovering Geometry Practice Your Skills 008 Kendall Hunt Publishing
7 4. It is too late to change the area. The length of the diagonals determines the area. LESSON 8.4 reas of Regular Polygons cm. a 7.8 cm. p 4.6 cm 4. n s 4 cm, a.8 cm, 8 cm cm 1.4 cm 7. r 10 cm r 7 cm LESSON 8.7 Surface rea cm. 40 cm cm cm cm cm cm cm 9. 1 sheet: front rectangle: ; back rectangle: ; bottom rectangle: 6; side trapezoids: 8; total 6. rea of 1 sheet 4 8. Possible pattern: 6. Possible answer (s will vary): s.1 cm, a.7 cm, 45.9 cm 1 Side Front 1 Le over Side 1 ack ottom LESSON 9.1 The Theorem of Pythagoras 7. pproimately 1.5 cm : area of square 6; area of square within angle ; area of octagon 10; area of octagon within angle ; shaded area cm LESSON 8.5 reas of ircles cm cm. cm 4. 4 cm cm cm cm cm cm or.6 cm 11. (64 18) square units 1. 5 cm LESSON 8.6 ny Way You Slice It cm 6.54 cm. cm.51 cm. 1 cm 7.70 cm 4. (16 ) cm 18.7 cm cm 4.41 cm 1. a 1 cm. p.9 cm h 14. in. 5. rea (11, 1); r 5 7. rea 49.7 cm 8. RV 15.4 cm 9. If the base area is 16 cm, then the radius is 4 cm. The radius is a leg of the right triangle; the slant height is the hypotenuse. The leg cannot be longer than the hypotenuse. 10. rea 150 in ; hypotenuse QR 5 in.; altitude to the hypotenuse 1 in. LESSON 9. The onverse of the Pythagorean Theorem 1. No. Yes. Yes 4. Yes 5. rea 1. cm 6. The top triangle is equilateral, so half its side length is.5. triangle with sides.5, 6, and 6.5 is a right triangle because So, the angle marked 95 should be cm. y the onverse of the Pythagorean Theorem, D is a right triangle, and D is a right angle. D and D are supplementary, so D is also a right triangle. Use the Pythagorean Theorem to find. Discovering Geometry Practice Your Skills NSWERS Kendall Hunt Publishing
8 cm 9. No. ecause, of is not a right angle. 10. annot be determined. The length of D is unknown. One possible quadrilateral is shown. 11. Yes. Using SSS, D D D. That means that the four angles of the quadrilateral are all congruent by PT. ecause the four angles must sum to 60 and they are all congruent, they must be right angles. So, D is a rectangle. LESSON 9. Two Special Right Triangles 1. a 14 cm. a 1 cm, b 4 cm. a 1 cm, b 6 cm cm 5. Perimeter 6 6 cm; area cm 6. 0 cm; 0 0 cm; area cm cm 8. 1, 9. ( 6, 6) 10. Possible answer: LESSON 9.4 Story Problems D The foot is about 8.7 away from the base of the building. To lower it by, move the foot an additional. away from the base of the building.. bout 6.4 km 0.4 km linear feet of trim must be painted, or 4. feet. Two coats means of coverage. Just over 1 quarts of paint is needed. If Hans buys quarts, he would have almost 1 quart le. It is slightly cheaper to buy 1 gallon and have about 1 1 quarts le. The choice is one of money versus conserving. Students may notice that the eaves etend beyond the eterior walls of the house and adjust their answer accordingly in., in in., in in km 1.7 km.1 km 1. km 0.9 km. km LESSON 9.5 Distance in oordinate Geometry units. 0 units. 17 units 4. D is a rhombus: ll sides 4, slope 5, slope, 5 so is not a right angle, and D is not a square. 5. TUVW is an isosceles trapezoid: TU and VW have slope 1, so they are parallel. UV and TW have length 0 and are not parallel (slope UV 1, slope TW ). 6. Isosceles; perimeter units 7. M(7, 10); N(10, 14); slope MN 4 ; slope 4 ; MN 5; 10; the slopes are equal; MN ( 1) (y 5) 4 9. enter (0, ), r The distances from the center to the three points on the circle are not all the same: P 61, P 61, P NSWERS Discovering Geometry Practice Your Skills 008 Kendall Hunt Publishing
9 LESSON 9.6 ircles and the Pythagorean Theorem 1. (5 4) cm, or about 54.5 cm. (7 4 ) cm, or about 49. cm. ( 58 7) cm 6.1 cm 4. rea cm cm 5. D cm 10.7 cm 6. ST LESSON 10.1 The Geometry of Solids 1. oblique. the ais. the altitude 4. bases 5. a radius 6. right 7. ircle or 10. or 11. Right pentagonal prism 1. DE and FGHIJ 1. F, G, H, DI, EJ 14. ny of F, G, H, DI, EJ or their lengths 15. False. The ais is not perpendicular to the base in an oblique cylinder. 16. False. rectangular prism has si faces. Four are called lateral faces and two are called bases. 17. True LESSON 10. Volume of Pyramids and ones cm cm cm 4. V V 8 a b 6. V 4 y 7. : 18 cubic units, : 144 cubic units. is larger. 8. : 5 cubic units, : 5 cubic units. They have equal volumes. 9. : 9 cubic units, : 7 cubic units. is larger. LESSON 10.4 Volume Problems cm. 178 in. 4 cans; 58 in.07 ; 4.6% lb (about 1 ton) 5. Note that E and E. V 8 cm ; S (8 4 ) cm 1.7 cm 6. bout 110,447 gallons truckloads LESSON 10.5 Displacement and Density ll answers are approimate cm. 7.8 g/cm g/cm in. 5. No, it s not gold (or at least not pure gold). The mass of the nugget is 165 g, and the volume is cm, so the density is 9.4 g/cm. Pure gold has density 19. g/cm. LESSON 10.6 Volume of a Sphere cm, or about cm. 18 cm, or about 56.5 cm LESSON 10. Volume of Prisms and ylinders cm. 144 cm cm 4. V 4y( ), or 8 y 1y 5. V 1 4 p h 6. V 6 1 y cm, or about 6. cm 4. 8 cm, or about 9. cm 5. 4 cm, or about 157. cm cm, or about 18. cm cm in in in ; 47.6% LESSON 10.7 Surface rea of a Sphere 1. V cm ; S cm. V 184. cm ; S 16.4 cm Discovering Geometry Practice Your Skills NSWERS Kendall Hunt Publishing
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