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The Plane

  • Page ID
    221073
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    The Plane

     

    1. The Distance Formula

      Recall that for two points (a,b) and (c,d) in a plane that the distance is found by the formula

       
         

                 Distance = \( \sqrt{(c - a)^2 + (d - b)^2} \)



      Example 1

      Find the distance between the points

      (1,1) and (-4,3)


      Solution

      Distance = \( \sqrt{-4 - 1)^2 + (3 - 1)^2} \)
             \( = \sqrt{25 + 4} = \sqrt{29} \) 
        

    2. The Midpoint Formula

      For points (a,b) and (c,d) the midpoint of the line segment formed by these points has coordinates:

       
              Midpoint Formula

             \( M = (\frac{a+c}{2}, \frac{b+ d}{2})\)

       

      Example:

      Suppose that you have a boat at one side of the lake with coordinates (3,4) and your friend has a boat at the other side of the lake with coordinates (18,22).  If you want to meet half way, at what coordinates should you meet?

      Solution:

              M = ((3 + 18)/2,(4 + 22)/2) = (10.5,13)
       

      Exercises

      1. Show that the points (-5,14), (1,4), and (11,10) are vertices of an isosceles triangle.

      2. Show that the triangle with vertices (1,1), (-1,-1), and (\(\sqrt{3},-\sqrt{3}\)) are vertices of a right triangle.
         

    3. Graphing on a Calculator

      We will graph the equations:

      1. y = 2x - 3  (Use graph then y(x) =)

      2. y = 5x2 + 4

      3. y = |x + 1|  (To find absolute value, use catalog then hit enter)

      4. y = 2x + {-1,0,1,2,3,5} (find the curly braces "{" and "}" use the list feature)

     



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