3.5. Did you ever think about street names? How does a city or town decide what to. composite figures

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1 .5 Composite Figures on the Coordinate Plane Area and Perimeter of Composite Figures on the Coordinate Plane LEARNING GOALS In this lesson, ou will: Determine the perimeters and the areas of composite figures on a coordinate plane. Connect transformations of geometric figures with number sense and operations. Determine the perimeters and the areas of composite figures using transformations. KEY TERM composite figures Did ou ever think about street names? How does a cit or town decide what to name their streets? Some street names seem to be ver popular. In the United States, almost ever town has a Main Street. But in France, there is literall a Victor Hugo Street in ever town! Victor Hugo was a French writer. He is best known for writing the novels Les Miserables and Notre-Dame de Paris, better known as The Hunchback of Notre Dame in English. If ou were in charge of naming the streets in our town, what names would ou choose? Would ou honor an people with their own streets? 17.5 Area and Perimeter of Composite Figures on the Coordinate Plane 17

2 Problem 1 Students are given the graph of a composite figure and asked to determine the perimeter and area of the figure. Students will draw line segments on the figure to divide it into familiar polgons and work with those polgons. The do this activit twice, dividing the composite figure two different was and conclude the area and perimeter remain unaltered. Grouping Ask a student to read the definition and information aloud. Discuss as a class. Have students complete Questions 1 through 4 with a partner. Then have students share their responses as a class. PROBLEM 1 Breakin It Down Now that ou have determined the perimeters and the areas of various quadrilaterals and triangles, ou can use this knowledge to determine the perimeters and the areas of composite figures. A composite figure is a figure that is formed b combining different shapes. To determine the area of a composite figure, divide it into basic shapes. 1. A composite figure is graphed on the coordinate plane shown. 16 D C E F Determine the perimeter of the composite figure. Round to the nearest tenth if necessar. Calculate the length of each horizontal or vertical segment. AB 5 6 () 5 FG 5 () 5 6 CD 5 4 () 5 6 HJ 5 6 (1) 5 1 DE JA EF 5 () A G B J H 16 Guiding Questions for Share Phase, Questions 1 through 4 How would ou describe the orientation of this composite figure on the coordinate plane? How man sides are on this composite figure? What familiar polgons did ou divide the composite figure into? Is the Distance Formula needed to calculate the length of an sides of the composite figure? Wh or wh not? Calculate the lengths of the remaining segments. BC GH BC GH BC 5 16 GH 5 65 BC 5 16 GH 5 65 P 5 AB 1 BC 1 CD 1 DE 1 EF 1 FG 1 GH 1 HJ 1 JA units The perimeter of this figure is approimatel 74.7 units. Is the Pthagorean Theorem needed to calculate the length of an sides of the composite figure? Wh or wh not? Is there more than one wa to divide this composite figure into familiar polgons? How? Would transforming the composite figure be helpful? Wh or wh not? 1 Chapter Perimeter and Area of Geometric Figures on the Coordinate Plane

3 . Draw line segments on the composite figure to divide the figure. Determine the area of the composite figure. Round to the nearest tenth if necessar. I divided the figure into two triangles, a square, and a rectangle. 1 Area of left triangle 5 (10)(6) Area of right triangle 5 (7)(4) 5 14 Area of rectangle 5 14(7) 5 9 Area of square A square units The area of this figure is 17 square units. Remember to use all of our knowledge about distance, area, perimeter, transformations, and the Pthagorean Theorem to make our calculations more efficient!. Draw line segments on the composite figure to divide the figure differentl from how ou divided it in Question. Determine the area of the composite figure. Round to the nearest tenth if necessar C D E 4 0 A B J I 16 F 1 16 G H I drew a large rectangle around the entire figure. I divided the top region that was not part of the original figure into a triangle and a rectangle. I divided the bottom region that was not part of the original figure into a rectangle and a trapezoid. Area of large rectangle 1 5 1(17) 5 06 Area of top triangle 5 (10)(6) 5 0 Area of top rectangle 5 10() 5 0 Area of bottom rectangle 5 10(4) Area of bottom trapezoid 5 (6 1 1)(4) 5 Area of figure 5 06 ( ) 5 17 The area of the figure is 17 square units..5 Area and Perimeter of Composite Figures on the Coordinate Plane 19

4 4. How does the area in Question compare to the area in Question? Eplain our reasoning. The areas of the composite figure in Question and Question are equal because dividing the composite figure differentl does not alter the shape or the size of the figure. Problem Students analze a representation of France mapped onto a coordinate plane and answer questions associated with the problem situation. Grouping Have students complete Questions 1 through 4 with a partner. Then have students share their responses as a class. PROBLEM Is France Heagonal? 1. Draw a heagon to approimate the shape of France. Use the heagon for Questions and. meters Brest UNITED KINGDOM English Channel Cherbourg Nantes Dunkerque Rouen Lille PARIS Orleans BELGIUM Nanc Dijon Strasbourg GERMANY LUXEMBOURG SWITZERLAND Guiding Questions for Share Phase, Questions 1 through 4 What method did ou use to compute the approimate length of the coastline? What method did ou use to compute the approimate area? How was the population of France determined? Did ou use a conversion? How? Ba of Bisca SPAIN Bordeau ANDORRA Limoges Perpignan Toulouse Valence Marseille meters Lon Grenoble Toulon Nice Mediterranean Sea ITALY MDNACO 0 Chapter Perimeter and Area of Geometric Figures on the Coordinate Plane

5 . Which of the following statements is true? The coastline of France is greater than 5000 kilometers. The coastline of France is less than 5000 kilometers. The coastline of France is approimatel 5000 kilometers. Can ou divide the heagon into more than one shape? Calculations will var depending on the heagon drawn in Question 1. The coastline of France is approimatel 47 kilometers, so the coastline of France is less than 5000 kilometers.. Which of the following statements is true? The area of France is greater than 1,000,000 square kilometers. The area of France is less than 1,000,000 square kilometers. The area of France is approimatel 1,000,000 square kilometers. The area of France is approimatel 547,000 square kilometers, so the area of France is less than 1,000,000 kilometers. 4. If the population of France is approimatel 11.4 people per square mile, how man people live in the countr of France? Approimatel 547, , or 65,000,000 people live in the countr of France..5 Area and Perimeter of Composite Figures on the Coordinate Plane 1

6 Talk the Talk Students draw line segments on a composite figure drawn on a coordinate plane to divide the figure into familiar polgons two different was and compute the area using each method. Talk the Talk Draw line segments on the composite figure to divide the figure into familiar shapes two different was, and then determine the area of the composite figure each wa to show the area remains unchanged. Grouping 0 15 Have students complete the Talk the Talk with a partner. Then have students share their responses as a class There are man was the composite figure can be divided into shapes. Have students present at least four different was and give reasons which wa the find preferable. The should support their opinions b being able to eplain how the calculated the area in each solution. Remind students that methods can involve addition and/ or subtraction. Answers will var. I etended the lines to form a square. The area of the original figure is equal to the area of the square minus the areas of the two triangles. The area of the square is 0 1, or 900 square units. The area of each triangle is (10)(10), or 50 square units. The area of the figure is 900 ( ), or 00 square units. I could also draw two vertical segments to create two congruent trapezoids and a rectangle. 1 The area of each trapezoid is (0 1 0)(10), or 50 square units. The area of the rectangle is 10(0), or 00 square units. The area of the figure is , or 00 square units. The area is the same using each method. Be prepared to share our solutions and methods. Chapter Perimeter and Area of Geometric Figures on the Coordinate Plane

7 Check for Students Understanding 1. Divide this region into familiar polgons b connecting vertices to form one or more line segments. E (9, 5) D (0, 5) A (0, 0) F (9, 4) B (0, 1) C (0, 1). Determine the perimeter of this composite figure a 1 b 5 c (9) 1 (4) 5 (AF) (AF) AF 5 97 < The approimate perimeter is 7. units..5 Area and Perimeter of Composite Figures on the Coordinate Plane A

8 . Determine the area of this composite figure. Area of Trapezoid: A 5 1 (b 1 1 b )h 5 1 (1 1 )9 5 1 (0) Area of Rectangle: A 5 bh 5 (11)(17) 5 17 The area of the composite figure is square units. B Chapter Perimeter and Area of Geometric Figures on the Coordinate Plane

9 Chapter Summar KEY TERMS bases of a trapezoid (.4) legs of a trapezoid (.4) composite figure (.5).1 Determining the Perimeter and Area of Rectangles and Squares on the Coordinate Plane The perimeter or area of a rectangle can be calculated using the distance formula or b counting the units of the figure on the coordinate plane. When using the counting method, the units of the -ais and -ais must be considered to count accuratel. Eample Determine the perimeter and area of rectangle JKLM. 400 J K M L The coordinates for the vertices of rectangle JKLM are J(10, 50), K(60, 50), L(60, 50), and M(10, 50). Because the sides of the rectangle lie on grid lines, subtraction can be used to determine the lengths. JK 5 60 (10) KL 5 50 (50) A 5 bh P 5 JK 1 KL 1 LM 1 JM (00) 5 54,000 The area of rectangle JKLM is 54,000 square units. The perimeter of rectangle JKLM is 960 units.

10 .1 Using Transformations to Determine the Perimeter and Area of Geometric Figures If a rigid motion is performed on a geometric figure, not onl are the pre-image and the image congruent, but both the perimeter and area of the pre-image and the image are equal. Knowing this makes solving problems with geometric figures more efficient. Eample Determine the perimeter and area of rectangle ABCD. A 0 B D A9 C B9 D9 C The vertices of rectangle ABCD are A(0, 0), B(60, 0), C(60, 60), and D(0, 60). To translate point D to the origin, translate ABCD to the right 0 units and down 60 units. The vertices of rectangle A9B9C9D9 are A9(0, 0), B9(0, 0), C9(0, 0), and D9(0, 0). Because the sides of the rectangle lie on grid lines, subtraction can be used to determine the lengths. A9D C9D P 5 A9B9 1 B9C9 1 C9D9 1 A9D The perimeter of rectangle A9B9C9D9 and, therefore, the perimeter of rectangle ABCD, is 00 units. A 5 bh 5 0(0) The area of rectangle A9B9C9D9 and, therefore, the area of rectangle ABCD, is 1600 square units. 4 Chapter Perimeter and Area of Geometric Figures on the Coordinate Plane

11 .1 Determining the Effect of Proportional and Non-Proportional Change on Perimeter and Area of a Rectangle Proportional Change The perimeter of a rectangle with base b and height h will change b a factor of k, given that its original dimensions are multiplied b a factor of k. The area of a rectangle with base b and height h will change b a factor of k, given that its original dimensions are multiplied b a factor of k. Eample Original Rectangle Rectangle Formed b Doubling Dimensions Rectangle Formed b Tripling Dimensions Linear Dimensions b 5 5 in. h 5 4 in. b 5 10 in. h 5 in. b 5 15 in. h 5 1 in. Rectangle 1 Perimeter (in.) (5 1 4) 5 1 (10 1 ) 5 6 (15 1 1) 5 54 Area (in. ) 5(4) () (1) 5 10 Non-Proportional Change The perimeter of a rectangle whose dimensions change non-proportionall b (adding to or subtracting from the dimensions) will change b a factor of 4. When the dimensions of a rectangle change non-proportionall, the resulting area changes, but there is not a clear pattern of increase or decrease. Eample Rectangle 1 Linear Dimensions Original Rectangle b 5 5 in. h 5 4 in. Rectangle Formed b Adding Inches to Dimensions b 5 7 in. h 5 6 in. Rectangle Formed b Adding Inches to Dimensions b 5 in. h 5 7 in. Perimeter (in.) (5 1 4) 5 1 (7 1 6) 5 6 ( 1 7) 5 54 Area (in. ) 5(4) 5 0 7(6) 5 4 (7) 5 56 Chapter Summar 5

12 . Determining the Perimeter and Area of Triangles on the Coordinate Plane The formula for the area of a triangle is half the area of a rectangle. Therefore, the area of a triangle can be found b taking half of the product of the base and the height. The height of a triangle must alwas be perpendicular to the base. On the coordinate plane, the slope of the height is the negative reciprocal of the slope of the base. Eample Determine the perimeter and area of triangle JDL. J 6 P D L The vertices of triangle JDL are J(1, 6), D(7, 9), and L(, ). JD 5 ( 1 ) 1 ( 1 ) DL 5 ( 1 ) 1 ( 1 ) LJ 5 ( 1 ) 1 ( 1 ) 5 (7 1) 1 (9 6) 5 ( 7) 1 ( 9) 5 (1 ) 1 (6 ) (6) 5 (7) P 5 JD 1 DL 1 LJ The perimeter of triangle JDL is approimatel 0.4 units. 6 Chapter Perimeter and Area of Geometric Figures on the Coordinate Plane

13 To determine the area of the triangle, first determine the height of triangle JDL. Slope of JD : m Slope of PL : m 5 Equation of JD : ( 1 ) 5 m( 1 ) Equation of PL : ( 1 ) 5 m( 1 ) ( 1) 5 ( ) Intersection of JD and PL, or P: The coordinates of P are (5.4,.) Height of triangle JDL: PL 5 ( 1 ) 1 ( 1 ) 5 ( 5.4) 1 (.) 5 (.6) 1 (5.) (5.4) Area of triangle JDL: A 5 1 bh 5 1 (JD)(PL) 5 1 ( 5 )(. ) 5 1 ( 169 ) The area of triangle JDL is 19.5 square units. Chapter Summar 7

14 . Doubling the Area of a Triangle To double the area of a triangle, onl the length of the base or the height of the triangle need to be doubled. If both the length of the base and the height are doubled, the area will quadruple. Eample Double the area of triangle ABC b manipulating the height. C9 C 6 4 Area of ABC Area of ABC9 B A A 5 1 bh A 5 1 bh 5 1 (5)(4) 5 1 (5)() B doubling the height, the area of triangle ABC9 is double the area of triangle ABC.. Determining the Perimeter and Area of Parallelograms on the Coordinate Plane The formula for calculating the area of a parallelogram is the same as the formula for calculating the area of a rectangle: A 5 bh. The height of a parallelogram is the length of a perpendicular line segment from the base to a verte opposite the base. Eample Determine the perimeter and area of parallelogram WXYZ. 6 4 Z W A Y X Chapter Perimeter and Area of Geometric Figures on the Coordinate Plane

15 The vertices of parallelogram WXYZ are W(, 5), X(, ), Y(, 5), and Z(4, 7). WX 5 ( 1 ) 1 ( 1 ) YZ 5 ( 1 ) 1 ( 1 ) 5 ( ()) 1 ( (5)) 5 (4 ) 1 (7 (5)) (6) 1 () WZ 5 ( 1 ) 1 ( 1 ) XY 5 ( 1 ) 1 ( 1 ) 5 (4 ()) 1 (7 (5)) 5 ( ) 1 (5 ()) 5 (1) 1 () 5 (1) 1 () P 5 WX 1 XY 1 YZ 1 WZ The perimeter of parallelogram WXYZ is approimatel 17.1 units. To determine the area of parallelogram WXYZ, first calculate the height, AY. Slope of base WX : m (5) () Slope of height AY : m 5 Equation of base WX : ( 1 ) 5 m( 1 ) Equation of height AY : ( 1 ) 5 m( 1 ) ( ()) 5 1 ( ) ( (5)) 5 ( ) Intersection of WX and AY, or A: ( 1 1 ) Chapter Summar 9

16 The coordinates of point A are ( 1 1, 1 ). AY 5 ( 1 ) 1 ( 1 ) 5 ( 1 1 ) 1 ( 5 ( 1 )) 5 ( 1 ) 1 ( 1 1 ) 5.5 Area of parallelogram WXYZ: A 5 bh A 5 10 (.5 ) A 5 10 The area of parallelogram WXYZ is 10 square units.. Doubling the Area of a Parallelogram To double the area of a parallelogram, onl the length of the bases or the height of the parallelogram needs to be doubled. If both the length of the bases and the height are doubled, the area will quadruple. Eample Double the area of parallelogram PQRS b manipulating the length of the bases. P Q S R S9 R Area of PQRS Area of PQR9S9 A 5 bh A 5 bh 5 (6)() 5 (1)() B doubling the length of the bases, the area of parallelogram PQR9S9 is double the area of parallelogram PQRS. 0 Chapter Perimeter and Area of Geometric Figures on the Coordinate Plane

17 .4 Determining the Perimeter and Area of Trapezoids on the Coordinate Plane A trapezoid is a quadrilateral that has eactl one pair of parallel sides. The parallel sides are known as the bases of the trapezoid, and the non-parallel sides are called the legs of the trapezoid. The area of a trapezoid can be calculated b using the formula A 5 ( b 1 b 1 ) h, where b 1 and b are the bases of the trapezoid and h is a perpendicular segment that connects the two bases. Eample Determine the perimeter and area of trapezoid GAME. G 16 1 A E 1 16 M The coordinates of the vertices of trapezoid GAME are G(4, 1), A(, 1), M(, 0), and E(4, 6). GA 5 ( 1 ) 1 ( 1 ) ME 5 ( 1 ) 1 ( 1 ) 5 ( (4)) 1 (1 1) 5 ((4) ) 1 ((6) 0) (6) 5 (6) 1 (6) EG 5 1 (6) AM P 5 GA 1 AM 1 ME 1 EG The perimeter of trapezoid GAME is approimatel 5.0 units. Chapter Summar 1

18 The height of trapezoid GAME is 6 units. A 5 ( b 1 b 1 ) h ( ) (6) 5 10 The area of trapezoid GAME is 10 square units..5 Determining the Perimeter and Area of Composite Figures on the Coordinate Plane A composite figure is a figure that is formed b combining different shapes. The area of a composite figure can be calculated b drawing line segments on the figure to divide it into familiar shapes and determining the total area of those shapes. Eample Determine the perimeter and area of the composite figure. P 6 T S 4 H 6 4 B G 4 6 R The coordinates of the vertices of this composite figure are P(4, 9), T(, 6), S(5, 6), B(5, 1), R(, 5), G(, 5), and H(4, 1). TS 5, SB 5 5, RG 5 5, HP 5 PT 5 ( 1 ) 1 ( 1 ) BR 5 ( 1 ) 1 ( 1 ) GH 5 ( 1 ) 1 ( 1 ) 5 ( (4)) 1 (6 9) 5 ( 5) 1 (5 1) 5 (4 ()) 1 (1 (5)) () 5 () 1 (6) 5 () 1 (6) P 5 PT 1 TS 1 SB 1 BR 1 RG 1 GH 1 HP Chapter Perimeter and Area of Geometric Figures on the Coordinate Plane

19 The perimeter of the composite figure PTSBRGH is approimatel 40.4 units. The area of the figure is the sum of the triangle, rectangle, and trapezoid formed b the dotted lines. Area of triangle: Area of rectangle: Area of trapezoid: A 5 1 ( bh A 5 bh A 5 b 1 b 1 ) h 5 1 (6)() 5 9(5) ( ) (6) The area of composite figure: A The area of the composite figure PTSBRGH is 96 square units. Chapter Summar