Areas of Parallelograms and Triangles. To find the area of parallelograms and triangles
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1 10-1 reas of Parallelograms and Triangles ommon ore State Standards G-MG..1 Use geometric sapes, teir measures, and teir properties to descrie ojects. G-GPE..7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles. MP 3, MP 4, MP 5, MP 6 Ojective To find te area of parallelograms and triangles You can comine triangles to make just aout any sape! MTHEMTIL PRTIES stage is eing set up for a concert at te arena. Te stage is made up of locks wit tops tat are congruent rigt triangles. Te tops of two of te locks, wen put togeter, make an 8 ft-y- 8 ft square. Te and as requested tat te stage e arranged to form te sape of an arrow. Draw a diagram tat sows ow te stage could e laid out in te sape of an arrow wit an area of at least 1000 ft 2 ut no more tan 1400 ft 2. 8 ft 8 ft Essential Understanding You can find te area of a parallelogram or a triangle wen you know te lengt of its ase and its eigt. parallelogram wit te same ase and eigt as a rectangle as te same area as te rectangle. Lesson Vocaulary ase of a parallelogram altitude of a parallelogram eigt of a parallelogram ase of a triangle eigt of a triangle Teorem 10-1 rea of a Rectangle Te area of a rectangle is te product of its ase and eigt. = Teorem 10-2 rea of a Parallelogram Te area of a parallelogram is te product of a ase and te corresponding eigt. = ase of a parallelogram can e any one 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 ase 616 apter 10 rea
2 Wy aren t te sides of te parallelogram considered altitudes? ltitudes must e perpendicular to te ases. Unless te parallelogram is also a rectangle, te sides are not perpendicular to te ases. Prolem 1 Finding te rea of a Parallelogram Wat is te area of eac parallelogram? 4.5 in. 4 in. 4.6 cm 3.5 cm 5 in. 2 cm You are given eac eigt. oose te corresponding side to use as te ase. = = = 5(4) = 20 Sustitute for and. = 2(3.5) = 7 Te area is 20 in. 2. Te area is 7 cm Wat is te area of a parallelogram wit ase lengt 12 m and eigt 9 m? Wat does F represent? F is an altitude of te parallelogram wen D and are used as ases. Prolem 2 Finding a Missing Dimension For D, wat is DE to te nearest tent? First, find te area of D. Ten use te area formula a second time to find DE. = = 13(9) = 117 Use ase D and eigt F. Te area of D is 117 in. 2. = 13 in. F D E 9.4 in. 117 = 9.4(DE) Use ase and eigt DE. DE = DE is aout 12.4 in. 2. parallelogram as sides 15 cm and 18 cm. Te eigt corresponding to a 15-cm ase is 9 cm. Wat is te eigt corresponding to an 18-cm ase? You can rotate a triangle aout te midpoint of a side to form a parallelogram. M Te area of te triangle is alf te area of te parallelogram. Lesson 10-1 reas of Parallelograms and Triangles 617
3 Teorem 10-3 rea of a Triangle Te area of a triangle is alf te product of a ase and te corresponding eigt. = 1 2 ase of a triangle can e any of its sides. Te corresponding eigt is te lengt of te altitude to te line containing tat ase. Wy do you need to convert te ase and te eigt into inces? You must convert tem ot ecause you can only multiply measurements wit like units. Prolem 3 Finding te rea of a Triangle Sailing You want to make a triangular sail like te one at te rigt. How many square feet of material do you need? Step 1 onvert te dimensions of te sail to inces. (12 ft # 12 in. ) + 2 in. = 146 in. Use a conversion factor. 1 ft (13 ft # 12 in. ) + 4 in. = 160 in. 1 ft Step 2 Find te area of te triangle. = 1 2 = 1 2 (160)(146) Sustitute 160 for and 146 for. = 11,680 Simplify. 12 ft 2 in. 13 ft 4 in. Step 3 onvert 11,680 in. 2 to square feet. 11,680 in. 2 # 1 ft 12 in. # 1 ft 12 in. = ft2 You need ft2 of material. 3. Wat is te area of te triangle? 5 in. 1 ft 1 in. 1 ft How do you know te lengt of te ase of te triangle? Te lower part of te figure is a square. Te ase lengt of te triangle is te same as te ase lengt of te square. Prolem 4 Finding te rea of an Irregular Figure Wat is te area of te figure at te rigt? Find te area of eac part of te figure. triangle area = 1 2 = 1 2 (6)8 = 24 in.2 square area = = 6(6) = 36 in. 2 area of te figure = 24 in in. 2 = 60 in Reasoning Suppose te ase lengts of te square and triangle in te figure aove are douled to 12 in., ut te eigt of eac polygon remains te same. How is te area of te figure affected? 8 in. 6 in. 618 apter 10 rea
4 Lesson eck Do you know HOW? Find te area of eac parallelogram m 20 m 8 ft 8 ft Do you UNDERSTND? MTHEMTIL PRTIES 5. Vocaulary Does an altitude of a triangle ave to lie inside te triangle? Explain. 6. Writing How can you sow tat a parallelogram and a rectangle wit te same ases and eigts ave equal areas? Find te area of eac triangle cm 16 cm 8 in. 7. D is divided into two triangles along diagonal. If you know te area of te parallelogram, ow do you find te area of? D Practice and Prolem-Solving Exercises MTHEMTIL PRTIES Practice Find te area of eac parallelogram. See Prolem m 15 cm 12 cm 4.7 in. 20 cm 5.7 in. 5.8 m 6 in. 4 m Find te value of for eac parallelogram. See Prolem Find te area of eac triangle. See Prolem yd 6 yd 5.7 m 5 m 4 m 3 ft 4 m 3 m 7.5 yd 2 ft 2 ft Lesson 10-1 reas of Parallelograms and Triangles 619
5 17. Uran Design akery as a 50 ft-y-31 ft parking lot. Te four parking spaces are congruent parallelograms, te driving region is a rectangle, and te two areas for flowers are congruent triangles. See Prolem 4. a. Find te area of te paved surface y adding te areas of te driving region and te four parking spaces.. Descrie anoter metod for finding te area of te paved surface. c. Use your metod from part () to find te area. Ten compare answers from parts (a) and () to ceck your work. 10 ft 15 ft 31 ft pply 18. Te area of a parallelogram is 24 in. 2 and te eigt is 6 in. Find te lengt of te corresponding ase. 50 ft 19. Wat is te area of te figure at te rigt? 64 cm 2 88 cm 2 96 cm cm rigt isosceles triangle as area 98 cm 2. Find te lengt of eac leg. 8 cm 14 cm 21. lgera Te area of a triangle is 108 in. 2. ase and corresponding eigt are in te ratio 3 : 2. Find te lengt of te ase and te corresponding eigt. 8 cm 22. Tink out a Plan Ki used geometry software to create te figure at te rigt. Se constructed < > and a point not on < >.Ten se constructed line k parallel to < > troug point. Next, Ki constructed point D on line k as well as D and D. Se dragged point D along line k to manipulate D. How does te area of D cange? Explain. Wic dimensions of te triangle cange wen Ki drags point D? Do te lengts of D and D matter wen calculating area? 23. Open-Ended Using grap paper, draw an acute triangle, an otuse triangle, and a rigt triangle, eac wit area 12 units 2. k D Find te area of eac figure. 24. JF 25. DJ 26. DKJ 27. DKJ 28. DKF 29. J 30. trapezoid DJF y 4 F J K 2 D O x 31. Reasoning Suppose te eigt of a triangle is tripled. How does tis affect te area of te triangle? Explain. 620 apter 10 rea
6 For Exercises 32 35, (a) grap te lines and () find te area of te triangle enclosed y te lines. 32. y = x, x = 0, y = y = x + 2, y = 2, x = y = x + 3, y = 0, x = y = 3 4 x - 2, y = -2, x = Proaility Your friend drew tese tree figures on a grid. fly lands at random at a point on te grid. a. Writing Is te fly more likely to land on one of te figures or on te lank grid? Explain.. Suppose you know te fly lands on one of te figures. Is te fly more likely to land on one figure tan on anoter? Explain. oordinate Geometry Find te area of a polygon wit te given vertices. 37. (3, 9), (8, 9), (2, -3), D(-3, -3) 38. E(1, 1), F(4, 5), G(11, 5), H(8, 1) 39. D(0, 0), E(2, 4), F(6, 4), G(6, 0) 40. K(-7, -2), L(-7, 6), M(1, 6), N(7, -2) Find te area of eac figure ft cm cm 200 m 25 ft 120 m 25 ft 20 cm 60 m 40 m allenge History Te 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. = 1s(s a)(s )(s c), were s = 1 2 (a + + c) Use Heron s Formula and a calculator to find te area of eac triangle. Round your answer to te nearest wole numer. 44. a = 8 in., =, c = 10 in. 45. a = 15 m, = 17 m, c = 21 m 46. a. Use Heron s Formula to find te area of tis triangle.. Verify your answer to part (a) y using te formula = in. 12 in. Lesson 10-1 reas of Parallelograms and Triangles 621
7 Standardized Test Prep ST/T 47. Te lengts of te sides of a rigt triangle are 10 in., 24 in., and 26 in. Wat is te area of te triangle? 116 in in in in In quadrilateral D, D D. Wic type of quadrilateral could D never e classified as? square rectangle romus kite Sort Response 49. re te side lengts of XYZ possile? Explain. Y 6 X 11 4 Z PERFORMNE TSK pply Wat You ve Learned Look ack at te information given aout te target on page 613. Te diagram of te target is sown again elow, wit tree vertices of te regular octagon laeled,, and. P is drawn perpendicular to. MTHEMTIL PRTIES MP 5 P O a. Wat is te measure of? Justify your answer.. re te four red triangles congruent? Justify your answer. c. Wat are te measures of te angles of P? d. Use a trigonometric ratio to find P to te nearest undredt of an inc. e. Find to te nearest undredt of an inc. f. Use your results from parts (d) and (e) to find te area of. Round your answer to te nearest tent of a square inc. 622 apter 10 rea
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