The Coordinate Plane, Slope, and Midpoint

The coordinate plane is formed when two perpendicular number lines intersect at a point called the origin, forming four quadrants.

Example 1

In which quadrant would you find each point?

Image Long Description

1. ( 3 , − 4 ) open 3 comma negative 4 close Move 3 units right and 4 units down. The point is in Quadrant IV.
2. ( − 2 , − 5 ) open negative 2 comma negative 5 close Move 2 units left and 5 units down. The point is in Quadrant III.

To find the slope of a line on the coordinate plane, choose two points on the line and use the slope formula.

Example 2

Find the slope of each line.

1. 

   m = y 2 − y 1 x 2 − x 1   = 2 − ( − 2 ) − 2 − 3   = 4 − 5 or − 4 5 table with 3 rows and 2 columns , row1 column 1 , m , column 2 equals . fraction y sub 2 , minus , y sub 1 , over x sub 2 , minus , x sub 1 end fraction , row2 column 1 , , column 2 equals . fraction 2 minus . open , negative 2 , close , over negative 2 minus 3 end fraction , row3 column 1 , , column 2 equals , 4 over negative 5 , or minus , 4 fifths , end table

2. 

   m = y 2 − y 1 x 2 − x 1   = 2 − 0 − 1 − ( − 1 ) = 2 0 table with 2 rows and 2 columns , row1 column 1 , m , column 2 equals . fraction y sub 2 , minus , y sub 1 , over x sub 2 , minus , x sub 1 end fraction , row2 column 1 , , column 2 equals . fraction 2 minus 0 , over negative 1 minus . open , negative 1 , close end fraction . equals , 2 over 0 , end table

   Since you cannot divide by zero, this line has an undefined slope.

If ( x m , y m ) open , x sub m , comma , y sub m , close is the midpoint of the segment joining ( x 1 , y 1 ) open , x sub 1 , comma , y sub 1 , close and ( x 2 , y 2 ) , open , x sub 2 , comma , y sub 2 , close comma then x m = x 1 + x 2 2 x sub m , equals . fraction x sub 1 , plus , x sub 2 , over 2 end fraction and y m = y 1 + y 2 2 . y sub m , equals . fraction y sub 1 , plus , y sub 2 , over 2 end fraction . .

Example 3

Find the coordinates of the midpoint of the segment with endpoints ( − 2 , 5 ) open negative 2 comma 5 close and ( 6 , − 3 ) . open 6 comma negative 3 close .

− 2 + 6 2 = 2 fraction negative 2 plus 6 , over 2 end fraction . equals 2 and 5 + ( − 3 ) 2 = 1 fraction 5 plus . open , negative 3 , close , over 2 end fraction . equals 1 so the midpoint is (2, 1).

Exercises

In which quadrant would you find each point? Graph each point on a coordinate plane.

1. (3, 2)
2. ( − 4 , 3 ) negative 4 comma 3 close
3. ( 2 , − 3 ) open 2 comma negative 3 close
4. ( 4 , − 2 ) open 4 comma negative 2 close
5. ( − 4 , − 5 ) open negative 4 comma negative 5 close
6. ( − 1 , − 3 ) open negative 1 comma negative 3 close

Find the slope of each line.

7. 
8. 
9. 
10. the line containing ( − 3 , 4 ) negative 3 comma 4 close and ( 2 , − 6 ) open 2 comma negative 6 close
11. the line containing (25, 40) and (100, 55)

Find the midpoint of the segment with the given endpoints.

12. ( − 4 , 4 ) , ( 2 , − 5 ) negative 4 comma 4 close comma open 2 comma negative 5 close
13. (3, 3), ( 7 , − 6 ) open 7 comma negative 6 close
14. ( − 1 , − 8 ) , ( 0 , − 3 ) open negative 1 comma negative 8 close comma open 0 comma negative 3 close
15. (3, 4), ( 2 , − 6 ) open 2 comma negative 6 close

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