1.1 RECTANGULAR COORDINATES Objectives Plot points in the Cartesian plane. Use the distance formula to find the distance between two points. Use the midpoint formula to find the midpoint of a line segment. Use the Cartesian plane to model and solve real-life problems.
The Cartesian Plane * Draw the Cartesian plane (rectangular coordinate system) and label all the important features.
* Describe how to plot ordered pairs ( x, y ) in the coordinate plane.
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Ex) Draw a coordinate plane and plot the following ordered pairs.
( −1, 2 ) , ( 3, 4 ) , ( 0, 0 ) , ( 3, 0 ) , ( −2, − 3)
* Why is plotting ordered pairs on the coordinate plane beneficial?
Ex) The table shows the numbers N (in millions) of subscribers to a cellular telecommunication service in the United States from 2001 through 2010, where t represents the year. Sketch a scatter plot of the data. What appears to be the relationship between the two variables? Year, t
2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
Subscribers, N
128.4 140.8 158.7 182.1 207.9 233.0 255.4 270.3 290.9 311.0
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The Pythagorean Theorem and the Distance Formula * Use the Pythagorean Theorem to derive a formula that gives the distance between two points ( x1 , y1 ) and
( x2 , y2 ) in the coordinate plane.
Ex) Show that the points ( 2, 1) , ( 4, 0 ) , and
( 5, 7 )
are vertices of a right triangle.
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The Midpoint Formula * Use the distance formula to prove that the midpoint between two points ( x1 , y1 ) and ( x2 , y2 ) is given by:
x1 + x2 y1 + y2 , 2 2
See pg. 110
Ex) Find the midpoint of the line segment joining ( −5, − 3) and ( 9, 3) .
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Applications Ex) Draw a coordinate plane and a triangle with vertices at ( −1, 2 ) , (1, − 4 ) , and
( 2, 3) .
Then, shift the
triangle three units to the right and two units up and re-draw the triangle. What are the vertices of the shifted triangle? NOTE: The above shifts are called “TRANSLATIONS” and are considered “RIGID MOTIONS” since the size and shape of the triangle are unchanged (the triangles are congruent).
