Rectangular Coordinates

Transcription

1 Rectangular Coordinates MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011

2 Objectives In this lesson we will learn to: plot points in the Cartesian plane, use the Distance Formula to find the distance between two points, use the Midpoint Formula to find the coordinates of the midpoint of a line segment, use the coordinate plane to model and solve application-style problems.

3 Cartesian Plane French mathematician René Descartes ( ) realized that by intersecting two number lines at a right angle, that he could locate any point in a plane. Quadrant II, 10 Quadrant I, Origin s 5 Horizontal axis (x-axis) Vertical axis (y-axis) Origin Quadrants, Quadrant III 10, Quadrant IV

4 Ordered Pairs A point in the plane is represented by an ordered pair of the form (x, y). x and y are the coordinates of the point. x represents the directed distance from the y-axis. y represents the directed distance from the x-axis.

5 Plotting Points 10 Graph the following 5 ordered pairs: {( 3, 1), ( 1, 3), (0, 2), (1, 5), ( 2, 4)} s 5

6 Plotting Points 10 Graph the following ordered pairs: {( 3, 1), ( 1, 3), (0, 2), (1, 5), 5 ( 2, 4)}

7 Scatter Plots The relationships between two measured variables are often revealed when the measurements are plotted as points in the plane. This is called a scatter plot. Example Create a scatter plot of month and temperature for the following data. Month Temperature Month Temperature

8 Solution Temperature h

9 Pythagorean Theorem Pythagorean Theorem For a right triangle with hypotenuse of length c and sides of lengths a and b, then a 2 + b 2 = c 2. a c b

10 Distance Formula A consequence of the Pythagorean Theorem is the Distance Formula. Distance Formula The distance between the points with coordinates (x 1, y 1 ) and (x 2, y 2 ) in the plane is d = (x 1 x 2 ) 2 + (y 1 y 2 ) 2.

11 Justification y x 1,y 1 y 1 d y 2 y 1 x 2,y 2 y 2 x 2 x 1 x 1 x 2 x

12 Verifying a Right Triangle Show that the points with coordinates ( 1, 3), (3, 5), and (5, 1) are the vertices of a right triangle.

13 Verifying a Right Triangle Show that the points with coordinates ( 1, 3), (3, 5), and (5, 1) are the vertices of a right triangle. d 1 = d 2 = d 3 = ( 1 3) 2 + (3 5) 2 = = 20 (5 3) 2 + (1 5) 2 = = 20 (5 ( 1)) 2 + (1 3) 2 = = 40 d d 2 2 = = 40 = d 2 3

14 Midpoint Formula The coordinates of the midpoint of a line segment joining two points are the averages of the coordinates of the endpoints of the line segment. Midpoint Formula The midpoint of the line segment joining the points (x 1, y 1 ) and (x 2, y 2 ) is given by ( x1 + x 2 Midpoint =, y ) 1 + y

15 Example Find the coordinates of the midpoint of the line segment whose endpoints have coordinates ( 7, 4) and (2, 8).

16 Example Find the coordinates of the midpoint of the line segment whose endpoints have coordinates ( 7, 4) and (2, 8). ( Midpoint =, ) = ( 52 ) 2 2, 2

17 Application A soccer player passes the ball from a point that is 18 yards from the endline and 12 yards from the sideline. The pass is received by a player 42 yards from the endline and 50 yards from the sideline. How long is the pass?

18 Application A soccer player passes the ball from a point that is 18 yards from the endline and 12 yards from the sideline. The pass is received by a player 42 yards from the endline and 50 yards from the sideline. How long is the pass? We must find the distance between the points with coordinates (12, 18) and (50, 42).

19 Application A soccer player passes the ball from a point that is 18 yards from the endline and 12 yards from the sideline. The pass is received by a player 42 yards from the endline and 50 yards from the sideline. How long is the pass? We must find the distance between the points with coordinates (12, 18) and (50, 42). d = (50 12) 2 + (42 18) 2 = = yards

20 Application Use the Midpoint Formula to estimate the sales for the company in Year Sales (in millions) 2003 $ $4245

21 Application Use the Midpoint Formula to estimate the sales for the company in Year Sales (in millions) 2003 $ $4245 We must find the midpoint of the line segment whose endpoints are (2003, 2800) and (2007, 4245).

22 Application Use the Midpoint Formula to estimate the sales for the company in Year Sales (in millions) 2003 $ $4245 We must find the midpoint of the line segment whose endpoints are (2003, 2800) and (2007, 4245). ( Midpoint =, 2 ) = (2005, ) 2

23 Application Use the Midpoint Formula to estimate the sales for the company in Year Sales (in millions) 2003 $ $4245 We must find the midpoint of the line segment whose endpoints are (2003, 2800) and (2007, 4245). ( Midpoint =, 2 ) = (2005, ) 2 Thus the sales in 2005 were approximately $3, 522, 500, 000.

24 Translating Points A triangle in the plane has vertices at coordinates (5, 8), (3, 6), and (5, 2). Find the coordinates of the vertices if the triangle is shifted two units left and one unit up.

25 Translating Points A triangle in the plane has vertices at coordinates (5, 8), (3, 6), and (5, 2). Find the coordinates of the vertices if the triangle is shifted two units left and one unit up. Original Coordinates Translated Coordinates (5, 8) (5 2, 8 + 1) = (3, 9) (3, 6) (3 2, 6 + 1) = (1, 7) (5, 2) (5 2, 2 + 1) = (3, 3)

26 Solution x x

27 Homework Read Section 1.1. Exercises: 1, 5, 9, 13,..., 69, 73