Perimeter and Area in the Coordinate Plane

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1. Perimeter and Area in the Coordinate Plane COMMON CORE Learning Standard HSG-GPE.B.7 HSG-MG.A.1 LOOKING FOR STRUCTURE To be proficient in math, ou need to visualize single objects as being composed of more than one object.

1 1. Perimeter and Area in the Coordinate Plane COMMON CORE Learning Standard HSG-GPE.B.7 HSG-MG.A.1 LOOKING FOR STRUCTURE To be proficient in math, ou need to visualize single objects as being composed of more than one object. Essential Question How can ou find the perimeter and area of a polgon in a coordinate plane? Work with a partner. a. On a piece of centimeter graph paper, draw quadrilateral ABCD in a coordinate plane. Label the points A(1, ), B( 3, 1), C(0, 3), and D(, 0). b. Find the perimeter of quadrilateral ABCD. c. Are adjacent sides of quadrilateral ABCD perpendicular to each other? How can ou tell? d. What is the definition of a square? Is quadrilateral ABCD a square? Justif our answer. Find the area of quadrilateral ABCD. Work with a partner. a. Partition quadrilateral ABCD into four right triangles and one square, as shown. Find the coordinates of the vertices for the five smaller polgons. b. Find the areas of the five smaller polgons. Area of Triangle BPA: Area of Triangle AQD: Area of Triangle DRC: Area of Triangle CSB: Area of Square PQRS: Finding the Perimeter and Area of a Quadrilateral B( 3, 1) Finding the Area of a Polgon B( 3, 1) S R A(1, ) C(0, 3) A(1, ) P Q C(0, 3) D(, 0) D(, 0) c. Is the sum of the areas of the five smaller polgons equal to the area of quadrilateral ABCD? Justif our answer. Communicate Your Answer 3. How can ou find the perimeter and area of a polgon in a coordinate plane?. Repeat Eploration 1 for quadrilateral EFGH, where the coordinates of the vertices are E( 3, 6), F( 7, 3), G( 1, 5), and H(3, ). Section 1. Perimeter and Area in the Coordinate Plane 9

2 1. Lesson What You Will Learn Core Vocabular Previous polgon side verte n-gon conve concave Classif polgons. Find perimeters and areas of polgons in the coordinate plane. Classifing Polgons Core Concept Polgons In geometr, a figure that lies in a plane is called a plane figure. Recall that a polgon is a closed plane figure formed b three or more line segments called sides. Each side intersects eactl two sides, one at each verte, so that no two sides with a common verte are collinear. You can name a polgon b listing the vertices in consecutive order. side BC B C verte D A E polgon ABCDE D The number of sides determines the name of a polgon, as shown in the table. You can also name a polgon using the term n-gon, where n is the number of sides. For instance, a 1-gon is a polgon with 1 sides. Number of sides Tpe of polgon 3 Triangle Quadrilateral 5 Pentagon 6 Heagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 1 Dodecagon n n-gon interior conve polgon interior concave polgon Classifing Polgons A polgon is conve when no line that contains a side of the polgon contains a point in the interior of the polgon. A polgon that is not conve is concave. Classif each polgon b the number of sides. Tell whether it is conve or concave. a. b. SOLUTION a. The polgon has four sides. So, it is a quadrilateral. The polgon is concave. b. The polgon has si sides. So, it is a heagon. The polgon is conve. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Classif the polgon b the number of sides. Tell whether it is conve or concave Chapter 1 Basics of Geometr

3 Finding Perimeter and Area in the Coordinate Plane You can use the formulas given below and the Distance Formula to find the perimeters and areas of polgons in the coordinate plane. REMEMBER Perimeter has linear units, such as feet or meters. Area has square units, such as square feet or square meters. Perimeter and Area Triangle c h b a s Square Rectangle w P = a + b + c A = 1 bh P = s A = s P = + w A = w Finding Perimeter in the Coordinate Plane READING You can read the notation ABC as triangle A B C. A(, 3) C(, 3) B(3, 3) Find the perimeter of ABC with vertices A(, 3), B(3, 3), and C(, 3). SOLUTION Step 1 Draw the triangle in a coordinate plane. Then find the length of each side. Side AB AB = ( 1 ) + ( 1 ) Distance Formula 7.81 Side BC = [3 ( )] + ( 3 3) Substitute. = 5 + ( 6) Subtract. = 61 Simplif. Use a calculator. BC = 3 = 5 Ruler Postulate (Postulate 1.1) Side CA CA = 3 ( 3) = 6 Ruler Postulate (Postulate 1.1) Step Find the sum of the side lengths. AB + BC + CA = So, the perimeter of ABC is about units. Monitoring Progress Find the perimeter of the polgon with the given vertices. Help in English and Spanish at BigIdeasMath.com 3. D( 3, ), E(, ), F(, 3). G( 3, ), H(, ), J( 1, 3) 5. K( 1, 1), L(, 1), M(, ), N( 3, ) 6. Q(, 1), R(1, ), S(, 1), T( 1, ) Section 1. Perimeter and Area in the Coordinate Plane 31

4 Finding Area in the Coordinate Plane Find the area of DEF with vertices D(1, 3), E(, 3), and F(, 3). SOLUTION Step 1 Draw the triangle in a coordinate plane b plotting the vertices and connecting them. D(1, 3) F(, 3) E(, 3) Step Find the lengths of the base and height. Base The base is FE. Use the Ruler Postulate (Postulate 1.1) to find the length of FE. FE = ( ) Ruler Postulate (Postulate 1.1) = 8 Subtract. = 8 Simplif. So, the length of the base is 8 units. Height The height is the distance from point D to line segment FE. B counting grid lines, ou can determine that the height is 6 units. Step 3 Substitute the values for the base and height into the formula for the area of a triangle. A = 1 bh Write the formula for area of a triangle. = 1 ( 8)(6) Substitute. = Multipl. So, the area of DEF is square units. Monitoring Progress Find the area of the polgon with the given vertices. 7. G(, ), H(3, 1), J(, 1) 8. N( 1, 1), P(, 1), Q(, ), R( 1, ) 9. F(, 3), G(1, 3), H(1, 1), J(, 1) 10. K( 3, 3), L(3, 3), M(3, 1), N( 3, 1) Help in English and Spanish at BigIdeasMath.com 3 Chapter 1 Basics of Geometr

Modeling with Mathematics You are building a shed in our backard. The diagram shows the four vertices of the shed. Each unit in the coordinate plane represents 1 foot.

5 Modeling with Mathematics You are building a shed in our backard. The diagram shows the four vertices of the shed. Each unit in the coordinate plane represents 1 foot. Find the area of the floor of the shed. 8 6 G(, 7) H(8, 7) 1 ft K(, ) J(8, ) ft SOLUTION 1. Understand the Problem You are given the coordinates of a shed. You need to find the area of the floor of the shed.. Make a Plan The shed is rectangular, so use the coordinates to find the length and width of the shed. Then use a formula to find the area. 3. Solve the Problem Step 1 Find the length and width. Length GH = 8 = 6 Ruler Postulate (Postulate 1.1) Width GK = 7 = 5 Ruler Postulate (Postulate 1.1) The shed has a length of 6 feet and a width of 5 feet. Step Substitute the values for the length and width into the formula for the area of a rectangle. A = w Write the formula for area of a rectangle. = (6)(5) Substitute. = 30 Multipl. So, the area of the floor of the shed is 30 square feet. M(, ) N(6, ). Look Back Make sure our answer makes sense in the contet of the problem. Because ou are finding an area, our answer should be in square units. An answer of 30 square feet makes sense in the contet of the problem. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1 ft R(, -3) P(6, -3) 11. You are building a patio in our school s courtard. In the diagram at the left, the coordinates represent the four vertices of the patio. Each unit in the coordinate plane represents 1 foot. Find the area of the patio. 1 ft Section 1. Perimeter and Area in the Coordinate Plane 33

6 1. Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE The perimeter of a square with side length s is P =.. WRITING What formulas can ou use to find the area of a triangle in a coordinate plane? Monitoring Progress and Modeling with Mathematics In Eercises 3 6, classif the polgon b the number of sides. Tell whether it is conve or concave. (See Eample 1.) In Eercises 13 16, find the area of the polgon with the given vertices. (See Eample 3.) 13. E(3, 1), F(3, ), G(, ) 1. J( 3, ), K(, ), L(3, 3) 15. W(0, 0), X(0, 3), Y( 3, 3), Z( 3, 0) 16. N(, 1), P(3, 1), Q(3, 1), R(, 1) In Eercises 17, use the diagram. In Eercises 7 1, find the perimeter of the polgon with the given vertices. (See Eample.) A( 5, ) B(0, 3) 7. G(, ), H(, 3), J(, 3), K(, ) 8. Q( 3, ), R(1, ), S(1, ), T( 3, ) 9. U(, ), V(3, ), W(3, ) F(, 1) 6 C(, 1) 10. X( 1, 3), Y(3, 0), Z( 1, ) 11. L(1, ) N(, 0) E(, 3) D(, 5) Find the perimeter of CDE. 18. Find the perimeter of rectangle BCEF. P( 1, ) M(, 0) 19. Find the perimeter of ABF. 1. F(, ) A(0, ) 0. Find the perimeter of quadrilateral ABCD. 1. Find the area of CDE. E(, ) D(0, ) B(, 0) C(, ). Find the area of rectangle BCEF. 3. Find the area of ABF.. Find the area of quadrilateral ABCD. 3 Chapter 1 Basics of Geometr

7 ERROR ANALYSIS In Eercises 5 and 6, describe and correct the error in finding the perimeter or area of the polgon Determine which points are the remaining vertices of a rectangle with a perimeter of 1 units. A A(, ) and B(, 1) B C(, ) and D(, ) C E(, ) and F(, ) D G(, 0) and H(, 0) 9. USING STRUCTURE Use the diagram. 6. P = + w = () + (3) = 1 The perimeter is 1 units. B(5, 1) A(, 3) 6 C(1, 1) F(0, ) E(0, 0) J(0, ) G(, ) H(, 0) K(, ) L(, 0) a. Find the areas of square EFGH and square EJKL. What happens to the area when the perimeter of square EFGH is doubled? b. Is this true for ever square? Eplain. b = 5 1 = h = (5 ) + (1 3) = MODELING WITH MATHEMATICS You are growing zucchini plants in our garden. In the figure, the entire garden is rectangle QRST. Each unit in the coordinate plane represents 1 foot. (See Eample.) A = 1 bh 1 ()(.) =. The area is about. square units. 1 1 Q(1, 13) R(7, 13) In Eercises 7 and 8, use the diagram P(, 1) Q(, 1) 7. Determine which point is the remaining verte of a triangle with an area of square units. A R(, 0) B S(, 1) C T( 1, 0) D U(, ) 6 U(1, ) V(, ) T(1, 1) W(, 1) S(7, 1) a. Find the area of the garden. b. Zucchini plants require 9 square feet around each plant. How man zucchini plants can ou plant? c. You decide to use square TUVW to grow lettuce. You can plant four heads of lettuce per square foot. How man of each vegetable can ou plant? Eplain. Section 1. Perimeter and Area in the Coordinate Plane 35

8 31. MODELING WITH MATHEMATICS You are going for a hike in the woods. You hike to a waterfall that is miles east of where ou left our car. You then hike to a lookout point that is miles north of our car. From the lookout point, ou return to our car. a. Map out our route in a coordinate plane with our car at the origin. Let each unit in the coordinate plane represent 1 mile. Assume ou travel along straight paths. b. How far do ou travel during the entire hike? c. When ou leave the waterfall, ou decide to hike to an old wishing well before going to the lookout point. The wishing well is 3 miles north and miles west of the lookout point. How far do ou travel during the entire hike? 3. HOW DO YOU SEE IT? Without performing an calculations, determine whether the triangle or the rectangle has a greater area. Which one has a greater perimeter? Eplain our reasoning. 3. THOUGHT PROVOKING Your bedroom has an area of 350 square feet. You are remodeling to include an attached bathroom that has an area of 150 square feet. Draw a diagram of the remodeled bedroom and bathroom in a coordinate plane. 35. PROBLEM SOLVING Use the diagram. L(, ) P(, ) 1 M(, ) 3 N(, ) a. Find the perimeter and area of the square. b. Connect the midpoints of the sides of the given square to make a quadrilateral. Is this quadrilateral a square? Eplain our reasoning. c. Find the perimeter and area of the quadrilateral ou made in part (b). Compare this area to the area ou found in part (a). 36. MAKING AN ARGUMENT Your friend claims that a rectangle with the same perimeter as QRS will have the same area as the triangle. Is our friend correct? Eplain our reasoning. 33. MATHEMATICAL CONNECTIONS The lines 1 1 = 6, = 3 +, and 3 = + are the sides of a right triangle. a. Use slopes to determine which sides are perpendicular. b. Find the vertices of the triangle. c. Find the perimeter and area of the triangle. Maintaining Mathematical Proficienc Solve the equation. (Skills Review Handbook) Q(, 1) R(, ) S(, ) 37. REASONING Triangle ABC has a perimeter of 1 units. The vertices of the triangle are A(, ), B(, ), and C( 1, ). Find the value of. Reviewing what ou learned in previous grades and lessons = = 0. + = = = = 3. Use a compass and straightedge to construct a cop of the line segment. (Section 1.) X Y 36 Chapter 1 Basics of Geometr

(0, 2) y = x 1 2. y = x (2, 2) y = 2x + 2

(0, 2) y = x 1 2. y = x (2, 2) y = 2x + 2 .5 Equations of Parallel and Perpendicular Lines COMMON CORE Learning Standards HSG-GPE.B.5 HSG-GPE.B. Essential Question How can ou write an equation of a line that is parallel or perpendicular to a given