Rhombi and Squares. Recognize and apply the properties of rhombi. Recognize and apply the properties of squares.

Transcription

1 hombi and quares ecognize and apply the properties of rhombi. ecognize and apply the properties of squares. Vocabulary rhombus square can you ride a bicycle with square wheels? rofessor tan Wagon at Macalester ollege in t. aul, Minnesota, developed a bicycle with square wheels. here are two front wheels so the rider can balance without turning the handlebars. iding over a specially curved road ensures a smooth ride. OEIE OF HOMI square is a special type of parallelogram called a rhombus. rhombus is a quadrilateral with all four sides congruent. ll of the properties of parallelograms can be applied to rhombi. here are three other characteristics of rhombi described in the following theorems. heorem 8.15 he diagonals of a rhombus are perpendicular. Example hombus 8.16 If the diagonals of a parallelogram are If, then perpendicular, then the parallelogram is is a rhombus. a rhombus. (onverse of heorem 8.15) 8.17 Each diagonal of a rhombus bisects a pair of opposite angles. You will prove heorems 8.16 and 8.17 in Exercises 5 and 6, respectively. tudy ip roof ince a rhombus has four congruent sides, one diagonal separates the rhombus into two congruent isosceles triangles. rawing two diagonals separates the rhombus into four congruent right triangles. Example 1 Given: rove: roof: roof of heorem 8.15 is a rhombus. y the definition of a rhombus,. rhombus is a parallelogram and the diagonals of a parallelogram bisect each other, so bisects at. hus,. because congruence of segments is reflexive. hus, by. by. and also form a linear pair. wo congruent angles that form a linear pair are right angles. is a right angle, so by the definition of perpendicular lines. Lesson 8-5 hombi and quares 41

2 tudy ip eading Math he plural form of rhombus is rhombi, pronounced OM-bye. Example Measures of a hombus LGE Use rhombus and the given information to find the value of each variable. a. Find y if m y 1. m 90 he diagonals of a rhombus are perpendicular. 1 y 1 90 ubstitution y 11 dd 1 to each side. y 11 ake the square root of each side. he value of y can be 11 or 11. b. Find m if m 56. m m m 56 Opposite angles are congruent. ubstitution he diagonals of a rhombus bisect the angles. o, m is 1 (56) or 8. OEIE OF UE If a quadrilateral is both a rhombus and a rectangle, then it is a square. ll of the properties of parallelograms and rectangles can be applied to squares. Example quares OOINE GEOMEY etermine whether parallelogram is a rhombus, a rectangle, or a square. List all that apply. Explain. (, 1) y (1, ) Explore lan lot the vertices on a coordinate plane. If the diagonals are perpendicular, then is either a rhombus or a square. he diagonals of a rectangle are congruent. If the diagonals are congruent and perpendicular, then is a square. O ( 1, ) x (, 1) olve Examine Use the istance Formula to compare the lengths of the diagonals. [ )] ( ( 1 1) (1 ) 1 ( ) Use slope to determine whether the diagonals are perpendicular. slope of 1 ( 1) or 1 slope of 1 1 or ince the slope of is the negative reciprocal of the slope of, the diagonals are perpendicular. he lengths of and are the same so the diagonals are congruent. is a rhombus, a rectangle, and a square. You can verify that is a square by finding the measure and slope of each side. ll four sides are congruent and consecutive sides are perpendicular. 4 hapter 8 uadrilaterals

3 hombus 1 raw any segment. lace the compass point at, open to the width of, and draw an arc above. Label any point on the arc as. Using the same setting, place the compass at, and draw an arc to the right of. lace the compass at, and draw an arc to intersect the arc drawn from. Label the point of intersection. 4 Use a straightedge to draw,, and. onclusion: ince all of the sides are congruent, quadrilateral is a rhombus. Example 4 iagonals of a quare ELL he infield of a baseball diamond is a square, as shown at the right. Is the pitcher s mound located in the center of the infield? Explain. ince a square is a parallelogram, the diagonals bisect each other. ince a square is a rhombus, the diagonals are congruent. herefore, the distance from first base to third base is equal to the distance between home plate and second base. hus, the distance from home plate to the center of the infield is 17 feet 8 inches 11 divided by or 6 feet 7 inches. his distance is longer than the distance from 1 6 home plate to the pitcher s mound so the pitcher s mound is not located in the center of the field. It is about feet closer to home. rd 90 ft nd itcher 17 ft in ft 6 in. Home 1st If a quadrilateral is a rhombus or a square, then the following properties are true. tudy ip quare and hombus square is a rhombus, but a rhombus is not necessarily a square. hombi 1. rhombus has all the properties of a parallelogram.. ll sides are congruent.. iagonals are perpendicular. 4. iagonals bisect the angles of the rhombus. roperties of hombi and quares quares 1. square has all the properties of a parallelogram.. square has all the properties of a rectangle.. square has all the properties of a rhombus. Lesson 8-5 hombi and quares 4

4 oncept heck 1. raw a diagram to demonstrate the relationship among parallelograms, rectangles, rhombi, and squares.. OEN ENE raw a quadrilateral that has the characteristics of a rectangle, a rhombus, and a square.. Explain the difference between a square and a rectangle. Guided ractice LGE In rhombus, x and 5x. 4. Find x. 5. Find. 6. Find m E. 7. Find m if m 8.. E OOINE GEOMEY Given each set of vertices, determine whether MN is a rhombus, a rectangle, or a square. List all that apply. Explain your reasoning. 8. M(0, ), N(, 0), (0, ), (, 0) 9. M( 4, 0), N(, ), (, ), (1, 1) 10. OOF Write a two-column proof. Given: GH, H, GH, and G are isosceles. rove: GH is a rhombus. G H pplication 11. EMOELING he teiner family is remodeling their kitchen. Each side of the floor measures 10 feet. What other measurements should be made to determine whether the floor is a square? ractice and pply For Exercises ee Examples 4 1 Extra ractice ee page 770. In rhombus, m m and Find m. 1. Find m. 14. Find. 15. Find m. E LGE Use rhombus XYZW with m WYZ 5, VW, XV a, and ZV 5a Find m YZV. 17. Find m XYW. 18. Find XZ. 19. Find XW. X 6 V W Y 5 Z 44 hapter 8 uadrilaterals OOINE GEOMEY Given each set of vertices, determine whether EFGH is a rhombus, a rectangle, or a square. List all that apply. Explain your reasoning. 0. E(1, 10), F( 4, 0), G(7, ), H(1, 1) 1. E( 7, ), F(, ), G(1, 7), H( 4, 7). E(1, 5), F(6, 5), G(6, 10), H(1, 10). E(, 1), F( 4, ), G(1, 5), H(, 1)

5 ONUION onstruct each figure using a compass and ruler. 4. a square with one side centimeters long 5. a square with a diagonal 5 centimeters long Use the Venn diagram to determine whether each statement is always, sometimes, or never true. 6. parallelogram is a square. uadrilaterals arallelograms 7. square is a rhombus. 8. rectangle is a parallelogram. 9. rhombus is a rectangle. 0. rhombus is a square. 1. square is a rectangle. hombi quares ectangles. EIGN Otto rutscher designed the plant stand at the left in 190. he base is a square, and the base of each of the five boxes is also a square. uppose each smaller box is one half as wide as the base. Use the information at the left to find the dimensions of the base of one of the smaller boxes.. EIMEE he diagonals of a rhombus are 1 centimeters and 16 centimeters long. Find the perimeter of the rhombus. esign he plant stand is constructed from painted wood and metal. he overall dimensions are 6 1 inches tall by 15 4 inches wide. ource: 4. his piece of art is orthea ockburne s Egyptian ainting: cribe. he diagram shows three of the shapes shown in the piece. Use a ruler or a protractor to determine which type of quadrilateral is represented by each figure. E F H L G M OOF Write a paragraph proof for each theorem. 5. heorem heorem 8.17 UH For Exercises 7 and 8, use the diagram of the court for squash, a game similar to racquetball and tennis. 7. he diagram labels the diagonal as 11,665 millimeters. Is this correct? Explain mm 6400 mm 11,665 mm ervice oxes 8. he service boxes are squares. Find the length of the diagonal mm 1600 mm Lesson 8-5 hombi and quares 45

6 9. FLG tudy the flags shown below. Use a ruler and protractor to determine if any of the flags contain parallelograms, rectangles, rhombi, or squares. Flags he state of Ohio has the only state flag in the United tates that is not rectangular. ource: World lmanac OOF enmark Write a two-column proof. t. Vincent and he Grenadines rinidad and obago 40. Given: WZY WXY, WZY 41. Given: X X and XYZ are isosceles. X X rove: WXYZ is a rhombus. rove: is a rhombus. W X X Z Y 4. Given: LG M 4. Given: and V are GH is a parallelogram. rhombi. rove: GH is a rhombus. rove: is equilateral. L M G H V 44. IIL HINING he pattern at the right is a series of rhombi that continue to form a hexagon that increases in size. opy and complete the table. Hexagon Number of rhombi x 45. WIING IN MH nswer the question that was posed at the beginning of the lesson. How can you ride a bicycle with square wheels? Include the following in your answer: difference between squares and rhombi, and how nonsquare rhombus-shaped wheels would work with the curved road. 46 hapter 8 uadrilaterals

7 tandardized est ractice 46. oints,,, and are on a square. he area of the square is 6 square units. Which of the following statements is true? he perimeter of rectangle is greater than 4 units. he perimeter of rectangle is less than 4 units. he perimeter of rectangle is equal to 4 units. he perimeter of rectangle cannot be determined from the information given. 47. LGE For all integers x, let <x> 1 x. Which of the following has x the greatest value? <0> <1> <> <4> Maintain Your kills Mixed eview LGE Use rectangle LMN, parallelogram LM, and the given information to solve each problem. (Lesson 8-4) 48. If LN 10, L x 1, and x 1, find x. 49. If m L 110, find m LM. 50. If m MN 5, find m MN. 51. If M 6x, L x y, and N 14 x, find x and y. 5. If m LM m MN, find m L. N L M OOINE GEOMEY etermine whether the points are the vertices of a parallelogram. Use the method indicated. (Lesson 8-) 5. (0, ), (6, 4), (4, 0), (, ); istance Formula 54. F(1, 1), G( 4, 1), H(, 4), (, 1); istance Formula 55. (, 7), L(, ), M(1, 7), N(, 1); lope Formula 56. ( 4, 1), (, 5), (1, 7), (, ); lope Formula efer to. (Lesson 6-4) 57. If 16, 4, and 9, find. 58. If y, y, 1, and 16, solve for y. 59. If 15, 1, and 8, find. efer to the figure. (Lesson 4-6) 60. If G, name two congruent angles. 61. If H, name two congruent angles. 6. If F F, name two congruent segments. 6. If, name two congruent segments. H G F Getting eady for the Next Lesson EEUIIE ILL olve each equation. (o review solving equations, see pages 77 and 78.) (8x 6x 7) (7x x 1) (4x 6 x 1) (7x x ) Lesson 8-5 hombi and quares 47