Polygons in the Coordinate Plane

Transcription

1 . Polgons in the Coordinate Plane How can ou find the lengths of line segments in a coordinate plane? ACTIVITY: Finding Distances on a Map Work with a partner. The coordinate grid shows a portion of a cit. Each square on the grid represents one square mile. 0 0 a. A public librar is located at, ). Cit Hall is located at, ). Plot and label these points. b. How far is the public librar from Cit Hall? c. A stadium is located miles from the public librar. Give the coordinates of several possible locations of the stadium. Justif our answers b graphing. d. Connect the three locations of the public librar, Cit Hall, and the stadium using our answers in part c). What shapes are formed? ACTIVITY: Graphing Polgons Work with a partner. Plot and label each set of points in the coordinate plane. Then connect each set of points to form a polgon. COMMON CORE Geometr In this lesson, ou will draw polgons in the coordinate plane. find distances in the coordinate plane. solve real-life problems. Learning Standard.G. Rectangle: A, ), B, 0), C, 0), D, ) Triangle: E, ), F, ), G, ) 0 0 Chapter Areas of Polgons

2 ACTIVITY: Finding Distances in a Coordinate Plane Math Practice Repeat Calculations What calculations are repeated? How can ou use this information to write a rule about the length of a line segment? Work with a partner. a. Find the length of each horizontal line segment in Activit. b. STRUCTURE What relationship do ou notice between the lengths of the line segments in part a) and the coordinates of their endpoints? Eplain. c. Find the length of each vertical line segment in Activit. d. STRUCTURE What relationship do ou notice between the lengths of the line segments in part c) and the coordinates of their endpoints? Eplain. e. Plot and label the points below in the coordinate plane. Then connect each pair of points with a line segment. Use the relationships ou discovered in parts b) and d) above to find the length of each line segment. Show our work. S, ) and T, ) U, ) and V, 0) W 0, ) and X 0, 0) Y, ) and Z, ) f. Check our answers in part e) b counting grid lines.. IN YOUR OWN WORDS How can ou find the lengths of line segments in a coordinate plane? Give eamples to support our eplanation.. Do the methods ou used in Activit work for diagonal line segments? Eplain wh or wh not.. Use the Internet or some other reference to find an eample of how finding distances in a coordinate plane is helpful in each of the following careers. a. b. c. Archaeologist Surveor Pilot Use what ou learned about finding the lengths of line segments to complete Eercises on page. Section. Polgons in the Coordinate Plane

3 . Lesson Lesson Tutorials You can use ordered pairs to represent vertices of polgons. To draw a polgon in a coordinate plane, plot and connect the ordered pairs. EXAMPLE Drawing a Polgon in a Coordinate Plane The vertices of a quadrilateral are A, ), B, ), C, ), and D, ). Draw the quadrilateral in a coordinate plane. Stud Tip After ou plot the vertices, connect them in order to draw the polgon. B, ) C, ) A, ) D, ) Plot and label the vertices. Connect the points to form the quadrilateral. Eercises Draw the polgon with the given vertices in a coordinate plane.. A0, 0), B, ), C, ). W, ), X, ), Y, ), Z, ). F, ), G, ), H, ), J, ). P, ), Q, ), R, ), S, ), T, ) Finding Distances in the First Quadrant You can find the length of a horizontal or vertical line segment in a coordinate plane b using the coordinates of the endpoints. When the -coordinates are the same, the vertical distance between the points is the difference of the -coordinates. When the -coordinates are the same, the horizontal distance between the points is the difference of the -coordinates. Be sure to subtract the lesser coordinate from the greater coordinate. a, b) c, b) c a d, f ) f g d, g) Chapter Areas of Polgons

4 Stud Tip EXAMPLE You can also find the length using vertices H and J. You can find the width using vertices F and J. Finding a Perimeter The vertices of a rectangle are F, ), G, ), H, ), and J, ). Draw the rectangle in a coordinate plane and find its perimeter. Draw the rectangle and use the vertices to find its dimensions. The length is the horizontal distance between F, ) and G, ), which is the difference of the -coordinates. length = = units The width is the vertical distance between G, ) and H, ), which is the difference of the -coordinates. width = = units F, ) G, ) So, the perimeter of the rectangle is ) + ) = 0 units. J, ) H, ) EXAMPLE Common Error You can count grid lines to find the dimensions, but make sure ou consider the scale of the aes. Real-Life Application In a grid of the ehibits at a zoo, the vertices of the giraffe ehibit are E 0, 0), F 0, 0), G 00, 0), and H 0, 0). The coordinates are measured in feet. What is the area of the giraffe ehibit? Plot and connect the vertices using a coordinate grid to form a trapezoid. 00 E0, 0) F0, 0) Use the coordinates to find the lengths b 0 of the bases and the height. b = 0 0 = 0 b = 00 0 = 00 h = 0 0 = 0 Use the formula for the area of a trapezoid. A = 0)0 + 00) = 0)0) = 00 0 h The area of the giraffe ehibit is 00 square feet. 0 0 b H0, 0) G00, 0) Eercises. The vertices of a rectangle are J, ), K, ), L,.), and M,.). Find the perimeter and the area of the rectangle.. WHAT IF? In Eample, the giraffe ehibit is enlarged b moving verte F to 0, 0). How does this affect the area? Eplain. Section. Polgons in the Coordinate Plane

5 . Eercises Help with Homework. WRITING How can ou use a coordinate plane to draw a polgon?. WRITING How can ou find the perimeter of a rectangle in a coordinate plane? +-)= +-)= +-)= +-)= Plot and label each pair of points in a coordinate plane. Find the length of the line segment connecting the points.. C 0, ), D, ). K, ), L, ). Q, ), R, ) Draw the polgon with the given vertices in a coordinate plane.. A, ), B, ), C 0, 0). D ),, E, ), F, ). G ),, H ),, J, ), K, ). L, ), M, ), N, ), P, ) 0. Q 0, ), R0, ), S, ), T0, ), U, 0). V, ), W, ), X, ), Y 0, ), Z, 0) Find the perimeter and the area of the polgon with the given vertices.. C, ), D, ), E, ), F, ). J, ), K, ), L, ), M, ). N0, ), P, ), Q, ), R 0, ). S, 0), T, ), U, ), V, 0). ERROR ANALYSIS Describe and correct the error in drawing a triangle with vertices A, ), B, ), and C, ). K, ) L, ) J, ) M, ) A, ) B, ) C, ). TREE HOUSE You design a tree house using a coordinate plane. You plot the vertices of the floor at J, ), K, ), L, ), and M, ). The coordinates are measured in feet. a. What is the shape of the floor? b. What are the perimeter and the area of the floor? Chapter Areas of Polgons

6 OPEN-ENDED Draw a polgon with the given conditions in a coordinate plane.. a square with a perimeter of 0 units. a rectangle with a perimeter of units 0. a rectangle with an area of units. a triangle with an area of units. STRUCTURE The coordinate plane. BUS ROUTE Polgon JKLMNP represents shows three vertices of a parallelogram. a bus route. Each grid square represents Find two possible points that could square miles. What is the shortest represent the fourth verte. distance, in miles, from station P to station L using the bus route? Eplain our reasoning. X, ) Y, ) W, ) K J P N L M. CITY LIMITS In a topographical map of a cit, the vertices of the cit limits are A0, ), B, ), C, ), D,.), and E 0,.). The coordinates are measured in miles. What is the area of the cit?. BACKYARD The vertices of a backard are W 0, 0), X 0, 00), Y 0, 00), and Z 0, 0). The coordinates are measured in feet. The line segment XZ separates the backard into a lawn and a garden. The area of the lawn is greater than the area of the garden. How man times larger is the lawn than the garden?. Precision The vertices of a rectangle are, 0),, a),, a), and, 0). The vertices of a parallelogram are, 0),, b),, b), and, 0). The value of a is greater than the value of b. Which polgon has a greater area? Eplain our reasoning. Divide. Write the answer in simplest form. Section.) MULTIPLE CHOICE You are filling bottles from gallons of lemonade. How man bottles can ou fill when each bottle is of a gallon? Section.) A B C D Section. Polgons in the Coordinate Plane