MATHEMATICS                          ANALYTICAL GEOMETRY 
                          DISTANCE FORMULA: to find length or distance 
                                                                                                   
                                                   
=( − ) +( − )  
                                                                                             
                          A. TRIANGLES:  Distance formula is used to show                                                                                            
                                      PERIMETER: sum of all the sides                                                                                                
                                      Scalene triangle: 3 unequal sides                                                                                              
                                      Isosceles triangle: 2 equal sides                                                                                              
                                      Equilateral triangle: 3 equal sides                                                                                            
                                      Right angled triangle:  
 = 
 + 
                                                                               
                          EXAMPLE 1 : Use the distance formula to show that the triangle below is right angled                                                       
                                                                                         R(6;6) 
                                           P(-1;3) 
                           
                                                               Q(1;1) 
                          PR =                                                           RQ=                                                            PQ= 
                           
                           
                                                     
         
        

                          Pythagorus:               =  +   
                           
                           
                          EXAMPLE 2 : Use the distance formula to determine if the triangle below is right angled.  
                                                                                                     S(0; 2)                  ST = 
                           
                                                                                                                              TR =  
                           
                                                                R(-3; -1)                                       T(0; -1)      RS =  
                          What other conclusion can you make about the triangle?                                                
         
         

                                                                                                                               =  +   
                           
                 EXAMPLE 3 :  The vertices of triangle UNR are given  U(-6; 1)  N(1; 4)  R(-3; -6). Use the distance 
                 formula to determine the type of triangle. Is it also right angled?. Calculate the perimeter.         
                  
                  
                  
                  
                  
                  
                  
                  
                 EXAMPLE 4: The vertices of triangle ABC are given,  A(-8; 9)  N(-2; -1)  R(7; 3). Use the distance 
                 formula to determine the type of triangle. Calculate the perimeter. 
                  
                  
                  
                  
                  
                  
                  
                  
                 EXAMPLE 5: The vertices of triangle PQR are given,  P(3;  27)  Q(0; 0)  R(6; 0). Use the distance 
                                                                            √
                 formula to determine the type of triangle. Calculate the perimeter.          
                  
                  
                  
                  
                  
                  
                  
                 EXAMPLE 6 : Triangle ABC is an isosceles triangle with vertices A(-7; -2)  B(-1; Y)  C(5; -2) with AB 
                 equal to BC.  Find the coordinate of y.. 
                  
                  
                  
                  
                  
                 EXAMPLE 7: Triangle DEF is equilateral with vertices D(4; 0)  E(-6; 0)   F(-1; Y) . Find y. 
                  
                  
                  
                  
                  
                 B. QUADRILATERALS : Distance formula is used to show:                                       
                         PARALLELOGRAM                                                                       
                         RHOMBUS                                                                             
                         RECTANGLE                                                                           
                         SQUARE                                                                              
                         KITE                                                                                
                         TRAPEZIUM 
                 1) PARALLELOGRAM                          CHARACTERISTICS                                   
                                                           - opposite sides equal and parallel               
                                                           - diagonals not equal                             
                                                           - diagonals are cut in half (BISECT) at the midpoint 
                                                           - corner angles are NOT 90˚                       
                                                           - opposite angles are supplementary               
                                                           -sum of 4 corner angles = 360˚ 
                 2) RHOMBUS                                - 4 equal sides                                   
                                                           - opposite sides equal and parallel               
                                                           - diagonals not equal                             
                                                           - diagonals are cut in half (BISECT) at the midpoint at 90˚
                                                           - corner angles are NOT 90˚                       
                                                           - opposite angles are supplementary               
                                                           -  sum of 4 corner angles = 360˚ 
                  
                 3) RECTANGLE                              - opposite sides equal and parallel                 
                                                           - diagonals ARE EQUAL                               
                                                           - diagonals are cut in half (BISECT) at the midpoint 
                                                           - corner angles ARE 90˚                             
                                                           - opposite angles are supplementary                 
                                                           -  sum of 4 corner angles = 360˚ 
                 4) SQUARE                                 - 4 equal sides                                     
                                                           -  opposite sides equal and parallel                
                                                           - diagonals ARE EQUAL                               
                                                           - diagonals are cut in half (BISECT) at the midpoint at 90˚
                                                           - corner angles are 90˚                             
                                                           - diagonals bisect corner angles into 45˚ + 45˚     
                                                           - opposite angles are supplementary                 
                                                           - sum of 4 corner angles = 360˚ 
                 5) KITE                                   - ADJACENT sides are Equal                          
                                                           - diagonals are not equal                           
                                                           - the long diagonal bisects the short diagonal at its midpoint
                                                           - the long diagonal bisects the short diagonal at 90˚ 
                                                           - the long diagonal bisects its corner angles       
                                                           - the angles at the ends of the short diagonal are equal 
                                                           - sum of the corner angles = 360˚ 
                 6) TRAPEZIUM                              - 4 sides which are not equal  BUT                  
                                                           - ONE PAIR OF OPPOSITE SIDES ARE PARALLEL 
                  
                  
                  
                 NOTE: With the parallelogram, rectangle, rhombus and square, if you can show that 2 PAIRS OF 
                 OPPOSITE SIDES ARE EQUAL then the opposite sides are also PARALLEL. 
                 In each of the following questions below the four vertices of the quadrilateral are given. Draw a 
                 rough diagram and use the distance formula to determine the type of quadrilateral. 
                 EXAMPLE 1:     R(-1; 1)  A(4; 2)  C(2; -1)  E(-3; -2)