Measuring Length 11and Area

Transcription

1 Measuring Lengt 11and Area 11.1 Areas of Triangles and Parallelograms 11.2 Areas of Trapezoids, Romuses, and Kites 11.3 Perimeter and Area of Similar Figures 11.4 Circumference and Arc Lengt 11.5 Areas of Circles and Sectors 11.6 Areas of Regular Polygons 11.7 Use Geometric Proaility Before In previous capters, you learned te following skills, wic you ll use in Capter 11: applying properties of circles and polygons, using formulas, solving for lengts in rigt triangles, and using ratios and proportions. Prerequisite Skills VOCABULARY CHECK Give te indicated measure for (P. 1. Te radius 2. Te diameter 3. mcadb D C P B A SKILLS AND ALGEBRA CHECK 4. Use a formula to find te widt w of te rectangle tat as a perimeter of 24 centimeters and a lengt of 9 centimeters. (Review p. 49 for 11.1.) In n ABC, angle C is a rigt angle. Use te given information to find AC. (Review pp. 433, 457, 473 for 11.1, 11.6.) 5. AB 5 14, BC m A 5 358, AB m B 5 608, BC Wic special quadrilaterals ave diagonals tat isect eac oter? (Review pp. 533, 542 for 11.2.) 9. Use a proportion to find XY if nuvw, nxyz. (Review p. 372 for 11.3.) U V 5 8 W X Y 12 Z 718

2 Now In Capter 11, you will apply te ig ideas listed elow and reviewed in te Capter Summary on page 779. You will also use te key vocaulary listed elow. Big Ideas 1 Using area formulas for polygons 2 Relating lengt, perimeter, and area ratios in similar polygons 3 Comparing measures for parts of circles and te wole circle KEY VOCABULARY ases of a parallelogram, p. 720 eigt of a parallelogram, p. 720 eigt of a trapezoid, p. 730 circumference, p. 746 arc lengt, p. 747 sector of a circle, p. 756 center of a polygon, p. 762 radius of a polygon, p. 762 apotem of a polygon, p. 762 central angle of a regular polygon, p. 762 proaility, p. 771 geometric proaility, p. 771 Wy? You can apply formulas for perimeter, circumference, and area to find and compare measures. To find lengts along a running track, you can reak te track into straigt sides and semicircles. Geometry Te animation illustrated elow for Example 5 on page 749 elps you answer tis question: How far does a runner travel to go around a track? Your goal is to find te distances traveled y two runners in different track lanes. Coose te correct expressions to complete te equation. Geometry at classzone.com Oter animations for Capter 11: pages 720, 739, 759, 765, and

3 11.1 Areas of Triangles and Parallelograms Before You learned properties of triangles and parallelograms. Now You will find areas of triangles and parallelograms. Wy? So you can plan a jewelry making project, as in Ex. 44. Key Vocaulary ases of a parallelogram eigt of a parallelogram area, p. 49 perimeter, p. 49 POSTULATES POSTULATE 24 Area of a Square Postulate For Your Noteook Te area of a square is te square of te lengt of its side. s POSTULATE 25 Area Congruence Postulate A 5 s 2 If two polygons are congruent, ten tey ave te same area. POSTULATE 26 Area Addition Postulate Te area of a region is te sum of te areas of its nonoverlapping parts. RECTANGLES A rectangle tat is units y units can e split into p unit squares, so te area formula for a rectangle follows from Postulates 24 and 26. THEOREM For Your Noteook THEOREM 11.1 Area of a Rectangle Te area of a rectangle is te product of its ase and eigt. Justification: Ex. 46, p. 726 A 5 READ DIAGRAMS Te word ase can refer to a segment or to its lengt. Te segment used for te eigt must e perpendicular to te ases used. PARALLELOGRAMS Eiter pair of parallel sides can e used as te ases of a parallelogram. Te eigt is te perpendicular distance etween tese ases. If you transform a rectangle to form oter parallelograms wit te same ase and eigt, te area stays te same. ase eigt ase at classzone.com 720 Capter 11 Measuring Lengt and Area

4 READ VOCABULARY Te eigt of a triangle is te lengt of te altitude drawn to te given ase. THEOREMS THEOREM 11.2 Area of a Parallelogram Te area of a parallelogram is te product of a ase and its corresponding eigt. Justification: Ex. 42, p. 725 THEOREM 11.3 Area of a Triangle Te area of a triangle is one alf te product of a ase and its corresponding eigt. Justification: Ex. 43, p. 726 For Your Noteook A 5 A 5 } 1 2 RELATING AREA FORMULAS As illustrated elow, te area formula for a parallelogram is related to te formula for a rectangle, and te area formula for a triangle is related to te formula for a parallelogram. You will write a justification of tese relationsips in Exercises 42 and 43 on pages Area of ~ 5 Area of Rectangle Area of n 5 1 } 2 p Area of ~ E X A M P L E 1 Use a formula to find area Find te area of ~PQRS. Solution P R 4 12 T 8 Metod 1 Use } PS as te ase. Te ase is extended to measure te eigt RU. So, 5 6 and 5 8. P 6 S U Area 5 5 6(8) 5 48 square units Metod 2 Use } PQ as te ase. Ten te eigt is QT. So, 5 12 and 5 4. Area (4) 5 48 square units GUIDED PRACTICE for Example 1 Find te perimeter and area of te polygon Areas of Triangles and Parallelograms 721

5 E X A M P L E 2 Solve for unknown measures DRAW DIAGRAMS Note tat tere are oter ways you can draw te triangle descried in Example 2. ALGEBRA Te ase of a triangle is twice its eigt. Te area of te triangle is 36 square inces. Find te ase and eigt. Let represent te eigt of te triangle. Ten te ase is 2. A 5 1 } 2 Write formula } 1 (2)() 2 Sustitute 36 for A and 2 for Simplify. 6 5 Find positive square root of eac side. c Te eigt of te triangle is 6 inces, and te ase is 6p inces. E X A M P L E 3 Solve a multi-step prolem PAINTING You need to uy paint so tat you can paint te side of a arn. A gallon of paint covers 350 square feet. How many gallons sould you uy? ANOTHER WAY In Example 3, you ave a triangle, so you can also find x y using trigonometry or special rigt angles. Solution You can use a rigt triangle and a rectangle to approximate te area of te side of te arn. STEP 1 Find te lengt x of eac leg of te triangle x 2 1 x 2 Use Pytagorean Teorem x 2 Simplify. Ï } x Solve for te positive value of x. 18 ft 26 ft 26 ft 18 ft STEP 2 Find te approximate area of te side of te arn. Area 5 Area of rectangle 1 Area of triangle 5 26(18) 1 1 } 2 p F(Ï } 338 )(Ï } 338 )G ft 2 STEP 3 Determine ow many gallons of paint you need. 637 ft 2 p} 1 gal ø 1.82 gal Use unit analysis. 350 ft 2 c Round up so you will ave enoug paint. You need to uy 2 gallons of paint. GUIDED PRACTICE for Examples 2 and 3 4. A parallelogram as an area of 153 square inces and a eigt of 17 inces. Wat is te lengt of te ase? 5. WHAT IF? In Example 3, suppose tere is a 5 foot y 10 foot rectangular window on te side of te arn. Wat is te approximate area you need to paint? 722 Capter 11 Measuring Lengt and Area

6 11.1 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 7, 23, and 37 5 STANDARDIZED TEST PRACTICE Exs. 2, 21, 30, 39, and VOCABULARY Copy and complete: Eiter pair of parallel sides of a parallelogram can e called its?, and te perpendicular distance etween tese sides is called te?. 2. WRITING Wat are te two formulas you ave learned for te area of a rectangle? Explain wy tese formulas give te same results. EXAMPLE 1 FINDING AREA Find te area of te polygon. on p. 721 for Exs COMPARING METHODS Sow two different ways to calculate te area of parallelogram ABCD. Compare your results. A B 8 E D 10 C ERROR ANALYSIS Descrie and correct te error in finding te area of te parallelogram. 10. A 5 5 (6)(5) A 5 5 (7)(4) PYTHAGOREAN THEOREM Te lengts of te ypotenuse and one leg of a rigt triangle are given. Find te perimeter and area of te triangle. 12. Hypotenuse: 15 in.; leg: 12 in. 13. Hypotenuse: 34 ft; leg: 16 ft 14. Hypotenuse: 85 m; leg: 84 m 15. Hypotenuse: 29 cm; leg: 20 cm EXAMPLE 2 on p. 722 for Exs ALGEBRA Find te value of x. 16. A 5 36 in A ft A cm 2 x 12 ft 17 cm 12 in. x x 11.1 Areas of Triangles and Parallelograms 723

7 19. ALGEBRA Te area of a triangle is 4 square feet. Te eigt of te triangle is alf its ase. Find te ase and te eigt. 20. ALGEBRA Te area of a parallelogram is 507 square centimeters, and its eigt is tree times its ase. Find te ase and te eigt. 21. OPEN-ENDED MATH A polygon as an area of 80 square meters and a eigt of 10 meters. Make scale drawings of tree different triangles and tree different parallelograms tat matc tis description. Lael te ase and te eigt. EXAMPLE 3 FINDING AREA Find te area of te saded polygon. on p. 722 for Exs ft cm m 8 ft 13 cm 10 m 17 ft 9 cm 11 cm 16 m in m in. 26 m 5 in. 19 in. 40 m 20 m 8 in. COORDINATE GRAPHING Grap te points and connect tem to form a polygon. Find te area of te polygon. 28. A(3, 3), B(10, 3), C(8, 23), D(1, 23) 29. E(22, 22), F(5, 1), G(3, 22) 30. MULTIPLE CHOICE Wat is te area of te parallelogram sown at te rigt? A 8 ft 2 6 in. 2 B 1350 in. C 675 in. 2 D ft 2 2 ft 3 in. 4 ft 2 in. 31. TECHNOLOGY Use geometry drawing software to draw a line l and a line m parallel tol. Ten draw n ABC so tat C is on linel and } AB is on line m. Find te ase AB, te eigt CD, and te area of n ABC. Move point C to cange te sape of n ABC. Wat do you notice aout te ase, eigt, and area of n ABC? 32. USING TRIGONOMETRY In ~ABCD, ase AD is 15 and AB is 8. Wat are te eigt and area of ~ABCD if m DAB is 208? if m DAB is 508? 33. ALGEBRA Find te area of a rigt triangle wit side lengts 12 centimeters, 35 centimeters, and 37 centimeters. Ten find te lengt of te altitude drawn to te ypotenuse. 34. ALGEBRA Find te area of a triangle wit side lengts 5 feet, 5 feet, and 8 feet. Ten find te lengts of all tree altitudes of te triangle. 35. CHALLENGE Te vertices of quadrilateral ABCD are A(2, 22), B(6, 4), C(21, 5), and D(25, 2). Witout using te Distance Formula, find te area of ABCD. Sow your steps WORKED-OUT SOLUTIONS on p. WS1 5 STANDARDIZED TEST PRACTICE

8 PROBLEM SOLVING 36. SAILING Sails A and B are rigt triangles. Te lengts of te legs of Sail A are 65 feet and 35 feet. Te lengts of te legs of Sail B are 29.5 feet and 10.5 feet. Find te area of eac sail to te nearest square foot. Aout ow many times as great is te area of Sail A as te area of Sail B? A EXAMPLE 3 on p. 722 for Ex MOWING You can mow 10 square yards of grass in one minute. How long does it take you to mow a triangular plot wit eigt 25 yards and ase 24 yards? How long does it take you to mow a rectangular plot wit ase 24 yards and eigt 36 yards? B 38. CARPENTRY You are making a tale in te sape of a parallelogram to replace an old 24 inc y 15 inc rectangular tale. You want te areas of two tales to e equal. Te ase of te parallelogram is 20 inces. Wat sould te eigt e? 39. SHORT RESPONSE A 4 inc square is a square tat as a side lengt of 4 inces. Does a 4 inc square ave an area of 4 square inces? If not, wat size square does ave an area of 4 square inces? Explain. 40. PAINTING You are earning money y painting a sed. You plan to paint two sides of te sed today. Eac of te two sides as te dimensions sown at te rigt. You can paint 200 square feet per our, and you carge $20 per our. How muc will you get paid for painting tose two sides of te sed? ft 6.5 ft 41. ENVELOPES Te pattern elow sows ow to make an envelope to fit a card tat is 17 centimeters y 14 centimeters. Wat are te dimensions of te rectangle you need to start wit? Wat is te area of te paper tat is actually used in te envelope? of te paper tat is cut off? 14 cm 14 cm 17 cm 42. JUSTIFYING THEOREM 11.2 You can use te area formula for a rectangle to justify te area formula for a parallelogram. First draw ~PQRS wit ase and eigt, as sown. Ten draw a segment perpendicular to ] PS troug point R. Lael point V. a. In te diagram, explain ow you know tat npqt > nsrv.. Explain ow you know tat te area of PQRS is equal to te area of QRVT. How do you know tat Area of PQRS 5? 17 cm P R P T S V 11.1 Areas of Triangles and Parallelograms 725

9 43. JUSTIFYING THEOREM 11.3 You can use te area formula for a parallelogram to justify te area formula for a triangle. Start wit two congruent triangles wit ase and eigt. Place and lael tem as sown. Explain ow you know tat Y Z XYZW is a parallelogram and tat Area of nxyw 5 1 } 2. X W 44. MULTI-STEP PROBLEM You ave enoug silver to make a pendant wit an area of 4 square centimeters. Te pendant will e an equilateral triangle. Let s e te side lengt of te triangle. a. Find te eigt of te triangle in terms of s. Ten write a formula for te area of te triangle in terms of s.. Find te side lengt of te triangle. Round to te nearest centimeter. 45. EXTENDED RESPONSE Te ase of a parallelogram is 7 feet and te eigt is 3 feet. Explain wy te perimeter cannot e determined from te given information. Is tere a least possile perimeter for te parallelogram? Is tere a greatest possile perimeter? Explain. 46. JUSTIFYING THEOREM 11.1 You can use te diagram to sow tat te area of a rectangle is te product of its ase and eigt. a. Figures MRVU and VSPT are congruent rectangles wit ase and eigt. Explain wy RNSV, UVTQ, and MNPQ are squares. Write expressions in terms of and for te areas of te squares.. Let A e te area of MRVU. Sustitute A and te expressions from part (a) into te equation elow. Solve to find an expression for A. Area of MNPQ 5 Area of MRVU 1 Area of UVTQ 1 Area of RNSV 1 Area of VSPT 47. CHALLENGE An equation of ] AB is y 5 x. An equation of ] AC is y 5 2. Suppose ] BC is placed so tat n ABC is isosceles wit an area of 4 square units. Find two different lines tat fit tese conditions. Give an equation for eac line. Is tere anoter line tat could fit tis requirement for ] BC? Explain. y A y 5 x B M R N U V S P T P C y 5 2 x MIXED REVIEW PREVIEW Prepare for Lesson 11.2 in Exs Find te lengt of te midsegment } MN of te trapezoid. (p. 542) M N M N M Te coordinates of npqr are P(24, 1), Q(2, 5), and R(1, 24). Grap te image of te triangle after te translation. Use prime notation. (p. 572) 51. (x, y) (x 1 1, y 1 4) 52. (x, y) (x 1 3, y 2 5) 53. (x, y) (x 2 3, y 2 2) 54. (x, y) (x 2 2, y 1 3) 27 N 726 EXTRA PRACTICE for Lesson 11.1, p. 916 ONLINE QUIZ at classzone.com

10 Extension Use after Lesson 11.1 Key Vocaulary unit of measure greatest possile error relative error Determine Precision and Accuracy GOAL Determine te precision and accuracy of measurements. All measurements are approximations. Te lengt of eac segment elow, to te nearest inc, is 2 inces. Te measurement is to te nearest inc, so te unit of measure is 1 inc. If you are told tat an oject is 2 inces long, you know tat its exact lengt is etween 1 1 } 2 inces and 2 1 } 2 inces, or witin 1 } 2 inc of 2 inces. Te greatest possile error of a measurement is equal to one alf of te unit of measure. Wen te unit of measure is smaller, te greatest possile error is smaller and te measurement is more precise. Using one-eigt inc as te unit of measure for te segments aove gives lengts of 1 6 } 8 inces and 2 3 } 8 inces and a greatest possile error of 1 } 16 inc. E X A M P L E 1 Find greatest possile error AMUSEMENT PARK Te final drop of a log flume ride is listed in te park guide as 52.3 feet. Find te unit of measure and te greatest possile error. Solution Te measurement 52.3 feet is given to te nearest tent of a foot. So, te unit of measure is 1 } 10 foot. Te greatest possile error is alf te unit of measure. Because 1 } } } , te greatest possile error is 0.05 foot. READ VOCABULARY Te precision of a measurement depends only on te unit of measure. Te accuracy of a measurement depends on ot te unit of measure and on te size of te oject eing measured. RELATIVE ERROR Te diameter of a icycle tire is 26 inces. Te diameter of a key ring is 1 inc. In eac case, te greatest possile error is 1 } 2 inc, ut a alf-inc error as a muc greater effect on te diameter of a smaller oject. greatest possile error Te relative error of a measurement is te ratio}}. measured lengt Rel. error 5 Bicycle tire diameter Key ring diameter 0.5 in. 0.5 in. } ø ø 1.9% Rel. error 5} % 26 in. 1 in. Te measurement wit te smaller relative error is said to e more accurate. Extension: Determine Precision and Accuracy 727

11 E X A M P L E 2 Find relative error PLAYING AREAS An air ockey tale is 3.7 feet wide. An ice rink is 85 feet wide. Find te relative error of eac measurement. Wic measurement is more accurate? Air ockey tale (3.7 feet) Ice rink (85 feet) Unit of measure 0.1 ft 1 ft Greatest possile error 1 } 2 p (unit of measure) 1 } 2 (0.1 ft) ft 1 } 2 (1 ft) ft Relative error greatest possile error }} measured lengt 0.05 ft } 3.7 ft ø ø 1.4% 0.5 ft } 85 ft ø ø 0.6% c Te ice rink widt as te smaller relative error, so it is more accurate. PRACTICE 1. VOCABULARY Descrie te difference etween te precision of a measurement and te accuracy of a measurement. Give an example tat illustrates te difference. EXAMPLE 1 on p. 727 for Exs. 2 5 EXAMPLE 2 on p. 728 for Exs. 6 9 GREATEST POSSIBLE ERROR Find te unit of measure. Ten find te greatest possile error in m km } 16 yd RELATIVE ERROR Find te relative error of te measurement cm in m mm 10. CHOOSING A UNIT You are estimating te amount of paper needed to make ook covers for your textooks. Wic unit of measure, 1 foot, 1 inc, or 1 } 16 inc, sould you use to measure your textooks? Explain. 11. REASONING Te greatest possile error of a measurement is 1 } 16 inc. Explain ow suc a measurement could e more accurate in one situation tan in anoter situation. PRECISION AND ACCURACY Tell wic measurement is more precise. Ten tell wic of te two measurements is more accurate cm; 12 cm ft; 25.6 ft in.; 13.4 ft ft; 35 in. 16. PERIMETER A side of te eraser sown is a parallelogram. Wat is te greatest possile error for te lengt of eac side of te parallelogram? for te perimeter of te parallelogram? Find te greatest and least possile perimeter of te parallelogram. 5.1 cm 1.4 cm 728 Capter 11 Measuring Lengt and Area