How To Understand The Theory Of Geometric Theory

Transcription

1 0-. Plan California Content Standards GEOM 8.0 Students know, derive, and solve prolems involving te perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures. GEOM 0.0 Students compute areas of polgons, including rectangles, scalene triangles, equilateral triangles, romi, parallelograms, and trapezoids. 0- Wat You ll Learn To find te area of a parallelogram To find te area of a triangle... nd W To find te force of wind against te side of a uilding, as in Example reas of Parallelograms and Triangles California Content Standards 8.0 Students know, derive, and solve prolems involving te area of common geometric figures. Introduce 0.0 Students compute areas of polgons, including rectangles, scalene triangles, and parallelograms. Introduce Lesson -9 Find te area of eac figure.. 5 cm. 8 in.. ft. a square wit 5-cm sides. a rectangle wit ase in. and eigt 7 in.. a. m--.5 m rectangle. a rectangle wit lengt ft and widt ft.5 m Eac rectangle is divided into two congruent triangles. Find te area of eac triangle. 8 units 5. units. units 7. California Mat Background New Vocaular ase of a parallelogram altitude of a parallelogram eigt of a parallelogram ase of a triangle eigt of a triangle Te area of, or numer of square units covered, a quadrilateral, can e descried algeraicall as te product of its ase and eigt. You can demonstrate tis transforming, cutting, and pasting sections of te quadrilateral to form a rectangle. Base and eigt also appear in te formulas for te area of a triangle (alf of a parallelogram) and te lateral surface areas of prisms and pramids. Because of tis, te area of a rectangle is often given in terms of ase and eigt, not lengt and widt. More Mat Background: p. 50C rea of a Parallelogram Te diagrams at te top of page 5 sow tat a parallelogram wit te same ase and eigt as a rectangle as te same area as te rectangle. Ke Concepts Teorem 0- rea of a Rectangle Te area of a rectangle is te product of its ase and eigt. = Teorem 0- rea of a Parallelogram Te area of a parallelogram is te product of a ase and te corresponding eigt. = Lesson Planning and Resources See p. 50E for a list of te resources tat support tis lesson. Bell Ringer Practice Vocaular Tip Te term ase is used to represent ot a segment and its lengt. ase of a parallelogram is an of its sides. Te corresponding altitude is a segment perpendicular to te line containing tat ase, drawn from te side opposite te ase. Te eigt is te lengt of an altitude. ltitude Base Ceck Skills You ll Need Use student page, transparenc, or PowerPoint. California Standards Dail Review Use transparenc Capter 0 rea Special Needs L Students ma not e familiar wit laeling te sides of a rectangle as eigt and ase. Point out tat a rectangle is a parallelogram wit four rigt angles, and te eigt of a rectangle is alwas equal to a side. Below Level L To elp students see tat CF is an altitude in Example, ave tem rotate te ook so tat D is orizontal and at te ottom of te parallelogram.

2 nline Visit: PHScool.com We Code: ce-0775 For: rea ctivit Use: Interactive Textook, 0- rea of a Triangle Finding te rea of a Parallelogram Find te area of eac parallelogram. a...5 in. in.. cm 5 in. cm You are given eac eigt. Coose te corresponding side to use as te ase. = = = 5() = 0 Sustitute. = (.5) = 7 Te area is 0 in.. Te area is 7 cm. Find te area of a parallelogram wit ase m and eigt 9 m. 08 m Finding a Missing Dimension For $BCD, find CF to te nearest tent. First, find te area of $BCD. Ten use te area formula a second time to find CF. = = 0() = 0 Use ase B and eigt DE. Te area of $BCD is 0 in.. = 0 = (CF) Use ase D and eigt CF. CF = 0 < 9. CF is aout 9. in. in. in. E 0 in. parallelogram as sides 5 cm and 8 cm. Te eigt corresponding to a 5-cm ase is 9 cm. Find te eigt corresponding to an 8-cm ase. 7.5 cm You can rotate a triangle aout te midpoint of a side to form a parallelogram..5 cm D F B C. Teac Guided Instruction Teacing Tip Make sure tat students understand tat 5 in. is te measure of te entire ase in part a. Error Prevention If students tink te ase tat corresponds to eigt CF is F rater tan D, remind tem tat te ase is alwas a side of te parallelogram. dditional Examples Find te area of te parallelogram. 9 m 8 m 0.5 m m parallelogram as 9-in. and 8-in. sides. Te eigt corresponding to te 9-in. ase is 5 in. Find te eigt corresponding to te 8-in. ase. 7.5 in. M Te area of te triangle is alf te area of te parallelogram. Ke Concepts Teorem 0- rea of a Triangle Te area of a triangle is alf te product of a ase and te corresponding eigt. = Lesson 0- reas of Parallelograms and Triangles 55 dvanced Learners L fter Examples and, ave students explore possile areas for a parallelogram wit side lengts cm and 0 cm. Te sould justif teir conclusions. Englis Learners EL Have students repeat aloud te formulas for te area of a rectangle, parallelogram, and triangle. Note tat in area equals one-alf ase times eigt, te ase and eigt must e perpendicular segments. 55

3 Guided Instruction Differentiating Instruction Have students cut out two copies of a triangle and ten join te triangles in tree was to form parallelograms. Students can readil see tat te area of eac parallelogram is twice te area of one triangle. Connection to lgera Have students carefull examine te formula. sk: Wat appens to force F if surface area doules? Force doules. Wat appens to force F if wind velocit v doules? Force quadruples. Find te area of XYZ. X Y dditional Examples cm 95 cm Te front of a garage is a square 5 ft on eac side wit a triangular roof aove te square. Te eigt of te triangular roof is 0. ft. To te nearest undred, ow muc force is exerted an 80 mi/ wind lowing directl against te front of te garage? Use te formula F = 0.00v. aout 7800 l Resources Dail Notetaking Guide 0- L Dail Notetaking Guide 0- dapted Version L Closure 0 cm cm n isosceles trapezoid as 0-m and 0-m ases, and te eigt is 8 m. Find te area. Hint: Draw a diagonal. 0 m Z Real-World In 99 tis uilding in Homestead, Florida, succumed to te 5 mi/ winds of Hurricane ndrew. Connection Example (page 55) 5 Capter 0 rea EXERCISES ase of a triangle is an of its sides. Te corresponding eigt is te lengt of te altitude to te line containing tat ase. Standards Practice Practice Example GO for Help Finding te rea of a Triangle Find te area of te triangle. = = (0)(.) = Sustitute and simplif. Te area of te triangle is ft. Find te area of te triangle. 0 cm Real-World Connection Structural Design Wen designing a uilding, ou must e sure tat te uilding can witstand urricane-force winds, wic ave a velocit of 7 mi/ or more. Te formula F = 0.00v gives te force F in pounds exerted a wind lowing against a flat surface. is te area of te surface in square feet, and v is te wind velocit in miles per our. How muc force is exerted a 7 mi/ wind lowing ft directl against te side of te uilding sown ere? Find te area of te side of te uilding. triangle area = = (0) = 0 ft ft rectangle area = = 0() = 0 ft 0 ft area of te side = = 00 ft Use te area of te side of te uilding and te velocit of te wind to find te force. F = 0.00v Use te formula for force. = 0.00(00)(7) Sustitute 00 for and 7 for v. = 9.8 Te force is aout 00 l, or. tons. Critical Tinking Suppose te ases of te rectangle and triangle in te uilding aove are douled to 0 ft, ut te eigt of eac figure remains te same. How is te force of te wind against te side of te uilding affected? Te force is douled. Find te area of eac parallelogram..5 m.79 in. 0. m... 5 cm cm 5.8 m 5.7 in..7 i 0 cm in. 5 cm 0 ft ft cm cm m. ft For more exercises, see Extra Skill, Word Prolem, and Proof Practice. 0 cm GEOM 8.0, 0.0 5

4 B Example (page 55) Example (page 5) Example (page 5) 0. Find te entire area of te lot and sutract te area for te flowers. 0c ft ppl Your Skills 5. Te area does not cange; te eigt and ase B do not cange. Visit: PHScool.com We Code: ce-00 x Find te value of for eac parallelogram Find te area of eac triangle m.5 d m d ft m m m.5 d 7.5 d 0. Landscaping Taisa s Baker as a plan for a 50 ft-- ft parking lot. Te four parking spaces are congruent parallelograms, te driving region is a rectangle, and te two unpaved areas 0 ft for flowers are congruent triangles. a. Find te area of te surface to e paved adding te areas of te driving region and 50 ft te four parking spaces. 90 ft. Descrie anoter metod for finding te area of te surface to e paved. c. Use our metod from part () to find 5 ft te area. Ten compare answers from ft parts (a) and () to ceck our work.. Te area of a parallelogram is in. and te eigt is in. Find te corresponding ase. in.. Multiple Coice Wat is te area of te figure cm at te rigt? B cm 9 cm 8 cm 88 cm cm. n isosceles rigt triangle as area of 98 cm. 8 cm Find te lengt of eac leg. cm. lgera In a triangle, a ase and a corresponding eigt are in te ratio :. Te area is 08 in.. Find te ase and te corresponding eigt. 8 in.; in. 5. Tecnolog Ki used geometr software to create te figure at te rigt. Se constructed C k D * ) and a point C not on * B ) B.Ten se constructed * ) line k parallel to B troug point C. Next, B Ki constructed point D on line k as well as D and BD. Se dragged point D along line k to manipulate #BD. How does te area of #BD cange? Explain. See left.. Open-Ended Using grap paper, draw an acute triangle, an otuse triangle, and a rigt triangle, eac wit area units. See margin. ft ft ft Lesson 0- reas of Parallelograms and Triangles 57. Practice ssignment Guide B -, -, -8 B 7-0, 7-, 9- C Callenge 7-9 Multiple Coice Practice 0, Mixed Review -5 Homework Quick Ceck To ceck students understanding of ke skills and concepts, go over Exercises, 0,, 9, 0. Exercises Make sure tat students also square te unit of measurement. Exercises 7 9 To elp students find corresponding eigts and ases, suggest tat te rotate teir textooks so tat eac parallelogram as a orizontal ase and vertical eigt. Exercise sk: How do ou know tat part of te figure is a square? four rigt angles and all sides are congruent Guided Prolem Solving Enricment Reteacing dapted Practice Practice Name Class Date Practice 0-. Coose te nets tat will fold to make a cue.. B. C. D. Draw a net for eac figure. Lael eac net wit its appropriate dimensions cm 8 cm cm cm cm 0 cm Matc eac tree-dimensional figure wit its net cm. B. C. D. Space Figures and Nets cm cm L L L L L. nswers ma var. Sample: 9. Coose te nets tat will fold to make a pramid wit a square ase.. B. C. D. Pearson Education, Inc. ll rigts reserved. Use Euler s Formula to find te missing numer. 0. Faces: 5. Faces: 7. Faces: 8 Edges: 7 Edges: 9 Edges: 8 Vertices: 5 Vertices: Vertices: 7 57

5 . ssess & Reteac. Find te area of te parallelogram. 5 ft 50 ft. Find te area of XYZW wit ase units and eigt units. square units. parallelogram as -cm and 8-cm sides. Te eigt corresponding to te 8-cm ase is.5 cm. Find te eigt corresponding to te -cm ase. cm. Find te area of RST. R 5 m Lesson Quiz 5 m 5. rectangular flag is divided into four regions its diagonals. Two of te regions are saded. Find te total area of te saded regions. 7 in. 87 in. 0 ft S ft m m in. lternative ssessment Have eac student draw and lael a triangle and a parallelogram, eac wit an area of 0 in. Have tem write a paragrap explaining ow te calculated te area of eac figure. T 7. 5 units 8. units 9. units 0. units. 7 units. units Exercise 8 C Callenge 58 Capter 0 rea GPS Find te area of eac figure. 7. $BJF 8. #BDJ 9. #DKJ 0. $BDKJ. $DKF. #BCJ. DJF units O B C D 8 0 x In Exercises 7, (a) grap te lines and () find te area of te triangle enclosed te lines. 7. See margin.. = x, x = 0, = 7 5. = x +, =, x =. =- x +, = 0, x =- 7. = x -, =-, x = 8. Find te area of te ellow triangular patc in te large field in te poto at te left. It as a ase of 0 d and a eigt of 0 d. 00 d 9. Proailit nn drew tese tree figures on a grid. fl lands at random at a point on te grid. a. Writing Is te fl more likel to land on one of te figures or on te lank grid? Explain. Blank grid; area is 8 units wile figures are units.. Suppose ou know te fl lands on one of te figures. Is te fl more likel to land on one figure tan on anoter? Explain. No; te figures ave te same area. Coordinate Geometr Find te area of a polgon wit te given vertices. 0 units 0. (, 9), B(8, 9), C(, -), D(-, -) 8 units. E(, ), F(, 5), G(, 5), H(8, ). D(0, 0), E(, ), F(, ), G(, 0). K(-7, -), L(-7, ), M(, ), N(7, -) 0 units 88 units Find te area of eac figure.. 5 ft 5. 5 cm. 00 m cm 5 ft 0 m 0 m 5 ft 0 cm 0 m.5 ft 55 cm,800 m Histor Te ancient Greek matematician Heron is most famous for tis formula for te area of a triangle in terms of te lengts of its sides a,, and c. = "s(s a)(s )(s c), were s = (a + + c) Use Heron s Formula and a calculator to find te area of eac triangle. Round our answer to te nearest wole numer. 7. a = 8 in., = 9 in., c = 0 in. in. 8. a = 5 m, = 7 m, c = m m 9. a. Use Heron s Formula to find te area of tis triangle.. Verif our answer to part (a) using te 5 in. 5 in. formula =. 5 in. 9 in. in. F J K. a 7. 5 units x 0 x 58 x O

6 Multiple Coice Practice For California Standards and CHSEE Tutorials, visit PHScool.com. We Code: cq-905 GEOM NP is te altitude to ML in te figure elow. Wat is te area of #MNL in square centimeters? C N California Resources Standards Master California Dail Review Transparencies California Review and Practice Workook 50 cm 7 cm 8 cm M P L CHSEE 7F.. Felicia orrowed mone from er moter to u a used car. Felicia agreed to pa er moter ack making equal-sized montl paments. fter paments, er alance was $5. fter 5 paments, it was $5775. Wic function can Felicia use to descrie te relationsip etween, te remaining alance, and p, te numer of paments se as made? D p = 5 75p = 9900p + 75 = p = Mixed Review Lesson 9-7 List te smmetries in eac tessellation.. See ack of ook. GO for Help D B Lesson -5 Te ase of te isosceles triangle is a side of a regular pentagon PENT. Find te measure of eac angle.. &PE &PN 7 E N T 5.. &PN 7 7. &PN 8. &EPN 9. &NT P G H Lesson See margin. Use a compass and straigtedge for te following constructions. * ) * ) 50. Draw a segment and lael it B. Construct D so tat D ' B at point. * ) * ) 5. Draw a segment. Lael it EF. Construct a line GH so tat GH EF. * ) 5. Draw a segment and lael it KL. Draw a point X not on * KL ). Construct a perpendicular from point X to KL (or to KL ). 5. E X F nline Visit: PHScool.com, We Code: ca K L 5. a.. a. 7. a. x O x x x O x 0 x O x x x 59