Quadrilaterals. Learning Objectives. Pre-Activity

Transcription

1 Section 3.4 Pre-Activity Preparation Quadrilateral Intereting geometric hape and pattern are all around u when we tart looking for them. Examine a row of fencing or the tiling deign at the wimming pool. Notice how quare, rectangle, parallelogram and other plane geometric figure combine to offer texture, interet, and important geometric tructure in our live. Modern and ancient deign in art make ue of imple geometric hape put together in complicated pattern. Tiling and moaic ue little quare of material to fit a pattern or make a picture. The architectural moaic on the outide of the Muhammad Ali Center in Louiville, Kentucky (at right) i contructed of ceramic tile rectangle of the ame ize but different color to form image of Muhammad Ali, the world-famou boxer. Modern encrypting oftware ue teellation and digital imaging to protect our privacy by liding and rotating regular haped polygon in a predictable pattern that can be coded or decoded. For more information look up the bold-faced word in any Internet earch engine. Learning Objective Find the perimeter of a quadrilateral Find the area of a quadrilateral Terminology Previouly Ued ide triangle New Term to Learn quadrilateral 07

2 08 Chapter 3 Geometry Building Mathematical Language Quadrilateral In the lat two ection, we ued formula for finding the perimeter and area of triangle and circle. Thi ection concern the next et of baic hape: quadrilateral cloed plane figure with four ide. cloed not cloed Following are profile of five baic quadrilateral hape, including their defining characteritic, perimeter, and area formula, and obervation or clarifying comment. Square A four-ided figure with all four ide equal. Each internal angle meaure 90 w l l w Rectangle A four-ided figure with oppoite ide equal and parallel. Each interior angle meaure 90 Perimeter (P) P = 4 Perimeter equal four time the length of one ide Area (A) A = Area equal ide quared OBSERVATIONS: Area meaurement ue quare unit one quare unit i a quare that i one unit (1 inch, foot, meter, mile, etc.) long on each ide: Perimeter (P) P = l + w or P = (l+w) Perimeter equal twice the length plu twice the width Area (A) A = lw Area equal length time width OBSERVATIONS: We have ued length multiplied by width to decribe multiplication the dimenional meaurement give context to finding the product. A w B h b b D w C Perimeter (P) P = b + w Perimeter equal twice the length (bae) plu twice the width Parallelogram A four-ided figure with oppoite ide equal and parallel. The interior angle are NOT necearily equal to 90 Area (A) A = bh Area equal bae time height OBSERVATIONS: There are three important meaurement for a parallelogram: bae (length), ide (width), and height (altitude). Height i required to find the area. h 3 (b) 1 (b1) 4 Perimeter (P) P = b 1 +b Perimeter equal the um of the length of each ide Trapezoid A four-ided figure with two unequal parallel ide and two non-parallel ide Area (A) A=½h(b 1 +b ) Area equal onehalf time the height time the um of the bae OBSERVATIONS: Many rooftop are trapezoidal in hape.

3 Section 3.4 Quadrilateral 09 B C Rhombu A h D A four-ided figure with all ide equal. It oppoite ide are parallel. Perimeter (P) P = 4 Perimeter equal four time the length of one ide Area (A) A = h alternatively: A = ½ d 1 d B C d d1 OBSERVATIONS: A rhombu i ometime called a diamond hape; think of a kite or a baeball infield. Can you ue the area formula for triangle to prove that: A D Area = 1 rhombu d d 1 given that the diagonal of a rhombu biect (cut in half) each other at right angle? Baeball Diamond or Baeball Square? Orientation i important in determining the name of a figure. Even though an infield i quare, the orientation make it look diamond haped. A quare i a rhombu whoe angle are each 90. Try it! Did You Know? A quadrilateral can alo be called a quadrangle. The meaning i till the ame: a figure with four angle and four traight ide. Many college have quadrangle at the center of their campue. Thi i Mob Quad at Merton College, Oxford, England.

4 10 Chapter 3 Geometry Methodologie Uing Geometric Formula Example 1: Find the area of a parallelogram that ha a bae of feet and a height of 18 inche. Example : Find the area of a quare with ide of 3 yard. Try It! Step in the Methodology Example 1 Example Step 1 Draw or examine a ketch of the information Step Determine which formula() to ue Step 3 Determine the unit needed Step 4 Make ure that all unit agree Make a ketch if neceary. When writing the formula, make ure that each part i identified with the information given. Sometime two or more formula will be needed to complete the information. Once the formula i choen, look back to determine what unit are required. Unit mut be the ame. Ue common converion ratio to change unit. Change the unit in the diagram if neceary. A = bh where 18 in ft b = the bae h= the height Area ue quare unit, o quare feet (ft ) or quare inche (in ) would be appropriate unit. We chooe to work in feet, o our anwer will be in quare feet. Unit are given in feet and inche. Ue a proportion equation to convert the unit: 18 in 1 in = x ft 1 ft 18 in : 1 ft x = = 1. 5 ft 1 in Replace 18 in with 1.5 ft.

5 Section 3.4 Quadrilateral 11 Step in the Methodology Example 1 Example Step 4 (con t) Make ure that all unit agree Unit mut be the ame. Ue common converion ratio to change unit. Change the unit in the diagram if neceary. Validate: Doe 1.5 ft = 18 in? 18 in 1 in =? 1. 5 ft 1 ft 18( 1) =? 1. 5( 1) 18 = 18 Step 5 Subtitute given meaurement into the formula Step 6 Solve Step 7 Validate: compare unit check computation Find needed information firt. Round each calculation to the deired number of decimal place. Make the calculation. Unit multiply like number. Two tep: firt compare calculated unit to the anticipated unit, then validate calculation. A good way to validate your calculation i to ubtitute your olution back into the formula and olve for one of the given value (or for the ingle given value, if there are only two variable in the formula). 1.5 ft ft b = ft, h = 1.5 ft A = ( ft)(1.5 ft) A = ( ft)(1.5 ft) A = ( 1.5) ft A = 3 ft ft wa anticipated Uing the calculated area and the given bae, olve for the height: A = bh, 3 ft = ft (h) 3 ft h = = 1. 5 ft ft

6 1 Chapter 3 Geometry Model Model 1: Perimeter Find the perimeter of a rectangular garden 7 meter wide and 1. meter long. Step 1 Step 7 m P = L + W 1. m Step 5 Step 6 P = L + W P = (1. m) + (7 m) P = 4.4 m + 14 m Anwer: P = 38.4 m Step 3 Step 4 Perimeter i meaured in linear unit. Anwer will be in meter. All neceary information i given in meter; unit agree. Step 7 Anwer in meter P = L + W 38.4 = L + (7) = L = L, L = 1. m Model : Area Find the area of a baeball infield meauring 90 feet between bae. Step 1 90 ft Step 5 A = (90 ft) Step 6 A = (90 ft)(90 ft) Anwer: Area = 8100 ft Step A = Step 3 quare unit (feet ) Step 4 Neceary unit are given in feet Step 7 quare feet A = 8100 = = 8100 = 90 ft

7 Section 3.4 Quadrilateral 13 Addreing Common Error Iue Incorrect Proce Reolution Correct Proce Validation Unit do not agree Find the area of a rectangular carpet runner that i 1 by 4. A = 1 4 = 88 In geometric formula, unit mut be the ame. 1 mut be changed to inche or 4 mut be changed to feet in order for the unit to agree. 4 in 1 ft = 144 in A = 4 in 144 in = 3456 in OR Area i quare unit; unit are inche o the anwer i quare inche in = l 144 in 3456 in = l 144 in l = 4 in 4 in = ft 1 ft A = ft 1 ft = 4 ft OR quare feet 4 ft = l 1 ft 4 ft = l 1 ft l = ft Uing the wrong unit How much carpet i needed for an area that meaure 1 by 10? Round up to the next whole quare yard. A = 1 10 = 10 ft Determine the unit requeted in the anwer before beginning your calculation. The unit are correct for area, but are not the requeted unit for the problem (quare yard). Convert feet to yard before proceeding. 1 ft 3 ft 1 =, x = = 4 yd x yd 1 yd 3 10 ft 3 ft 10 =, x = x yd 1 yd 3 10 A = 4 yd yd 3 40 = yd 3 yd quare yard yd ' yd = = yd ' yd = yd = 4 yd 3 10 Note: > 13 yard of carpet i needed, o get 14 yard. = yd. 14 yd rounded up to the next whole quare yard

8 14 Chapter 3 Geometry Iue Incorrect Proce Reolution Correct Proce Validation Not validating unit Find the area of a field that meaure 4 feet by 30 yard. A = lw A = 4 30 A = 160 ft Carrying unit along in calculation help validate that the work wa done correctly. 4 ft 3 ft = x yd 1 yd 4 x = = 14 yd 3 A = lw A = 14 yd 30 yd = 40 yd Area i q feet or q yard. The anwer i in quare yard. 40 yd = l : 30 yd 40 yd 30 yd 14 yd = l = l 14 yd 1 yd = x ft 3 ft x = 4 ft Incorrect drawing or ketch A parallelogram ha a bae of 11 inche, a width of 13 inche and a height of 1 inche. What i the perimeter? 11 in 1 in The height i the perpendicular ditance from the bae to the top of a figure. Be ure to check your drawing againt the information provided. 1 in 11 in P = b + w = (11) + (13) = + 6 = 48 in 13 in inche 48 = ( 11 in) + w 6 in = w w = 13 in P = b + w = (11) + (1) = + 4 = 46 in Uing an incorrect formula Find the perimeter of the parallelogram below: 8 m 10 m P = 4 = 4(10) = 40 m Be ure to verify what hape you re working with and that you are applying the correct formula. The quadrilateral i identified in the problem a a parallelogram (not a rhombu). The correct formula for finding the perimeter of a parallelogram i: P = b + w The correct calculation i: P = (10) + (8) = 36 m meter 36 m = (10 m) + w 16 m = w w = 8 m

9 Section 3.4 Quadrilateral 15 Preparation Inventory Before proceeding, you hould be able to ue the correct formula to calculate the following: Area and perimeter of a rectangle Area and perimeter of a trapezoid Area and perimeter of a parallelogram Square from a Parallelogram? The quare in thi drawing are all baed on the parallelogram. The top and bottom quare each have ide the ame length a the bae of the parallelogram. The left and right quare have ide the ame length a the ide of the parallelogram. When you draw a line from the center of each of the quare you get a new quare. Thi particular idea i baed on a problem poed by French mathemetician Victor Thébault. There are many more intereting geometric problem baed on quadrilateral. To learn more, try earching online.

10 Section 3.4 Activity Quadrilateral Performance Criteria Finding the perimeter and area of quadrilateral. ue of the appropriate formula accuracy of calculation validation of the anwer Critical Thinking Quetion 1. What are four application for area?. Why i perimeter meaured in linear unit? 3. Why doe area ue quare unit? 4. Why do unit have to be the ame in order to find perimeter or area? 16

11 Section 3.4 Quadrilateral What value doe a ketch provide for olving a geometric problem? 6. Why i the height ued in finding the area of parallelogram and trapezoid? 7. The formula for finding the area of a rectangle and the area of a parallelogram are very imilar. Why? Tip for Succe Good practice include validating by correctly identifying unit of meaure: linear unit for perimeter and quare unit for area Draw and label a diagram or ketch a accurately a poible ue graph paper a a tool to help you

12 18 Chapter 3 Geometry Demontrate Your Undertanding 1. Find the perimeter a indicated for each of the following: Problem Worked Solution Validation a) a rectangle with length 14 m and width 7 m b) a rectangle with length 3.5 feet and width 8 inche c) Meaurement of the roof are: top = 15 ft bottom = 0 ft ide = 10 ft height = 8 ft What i the length of a tring of light framing the front of the roof (the part viible in the illutration)? d) A paper kite in a rhombu hape ha ide of 30 in, a height of 1 in. The mall diagonal i 15 in and the large diagonal i 48 in. How much fringe i needed to go around the kite?

13 Section 3.4 Quadrilateral 19. Find the area a indicated for each of the following: Problem Worked Solution Validation a) a rectangle with length 4 mile and width.3 mile b) a rectangle with length of 5 inche and width of three feet c) Meaurement of the roof are: top = 0 ft bottom = 34 ft ide = 5 ft height = 4 ft Find the area of the front of the roof (the part viible in the illutration).

14 0 Chapter 3 Geometry Problem Worked Solution Validation d) A paper kite in a rhombu hape ha ide of 30 in and a height of 1 in. The mall diagonal i 15 in and the large diagonal i 48 in. How much paper wa ued to make the kite? (Ue the diagram below a needed.) h d d 1

15 Section 3.4 Quadrilateral 1 Identify and Correct the Error In the econd column, identify the error() in the worked olution or validate it anwer. If the worked olution i incorrect, olve the problem correctly in the third column and validate your anwer. Worked Solution Identify Error or Validate Correct Proce Validation 1) A cabinet door meaure 3 feet by 15 inche. What i the area of the door? 3 ft 15 in A = bh = 3(15) = 45 inche ) Find the area of a quare that meaure. yard on each ide.. yd Area = 4 = 4 (.) = 8.8 yd 3) Find the area of a parallelogram that ha adjacent ide of 3 feet and 7 feet and a height of feet. 3 7 A = bh = 7 ft ( ft) = 14 ft

16 Chapter 3 Geometry Worked Solution Identify Error or Validate Correct Proce Validation 4) Find the perimeter of the trapezoid hown below: 30 in 9 in ft 38 in 75 in P = b 1 + b = 75 in + 30 in + 9 in + 38 in = 17 in 5) Find the area of a rhombu-haped kite if each edge i 50 cm and the height i 46 cm A = h( b1 + b) 1 = 46 cm ( 50 cm + 50 cm) = 3 cm ( 100 cm) = 300 cm