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
   
   

   
   
   Example 1

   Find the distance between the points

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

          
   
   

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

   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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