What is the other endpoint of a line segment if the endpoint is (-9,-5) and the midpoint is (-1,1)?

Question

I can somehow never remember how to do this...

Answer 1

to find the midpoint, add the x endpoints and divide by 2 and add the y endpoints and divide by 2

 

(-9,-5)__________(-1,1)__________(x,y)

 

(-9+x)/2=-1

(-5+y)/2=1

 

-9+x=-2

x=9-2

x=7

 

-5+y=2

y=5+2

y=7

the other endpoint has coordinates (7,7)

 

Answer 2

 the distance is sqrt ( (-9 - -1)^2 + (-5 -1)^2 ) =  sqrt ( (-8)^2 + (-6)^2 ) = sqrt ( 64 + 36 ) = sqrt(100) = 10

 

  the slope is M = (-5 -1)/(-9 - -1) = -6/-8 = 3/4

 

Distance from (X,y) , the missing endpoint, to (-1,1) must also be 10.

 

(x+1)^2 + (y-1)^2 = 100  <--- equation #1

 

The slope must also be the same:

 

   (y-1)/(x_1) = 3/4  --->  4(y-1) = 3(x+1) 

                                 4y - 4 = 3x + 3

                              4y = 3x + 7

                             y = (3/4)x + 7/4

 

   (x+1)^2 + (7/4 + 3/4X - 1)^2 = 100

  (x+1)^2 + (3/4x + 3/4)^2 = 100

 

  (x+1)^2 + (3/4)^2(x+1)^2 = 100

 

  25/16 (x+1)^2 = 100

 

      (x+1)^2 = 1600/25

 

     (x+1)^2 = 64

 

     x+1 = +or- 8

 

   x = -1 +or- 8

 

  The one answer that makes sense is x = -1+8 = 7, as

  x is increasing with positive slope, based on the locations

   of the given endpoint and midpoint.'

 

Plugging into the second equation:

 

    y = 3/4(7)+3/4 = 21/4 + 7/4 = 28/4= 7

 

  So the other endpoint is (7,7)

 

CHECK:

 

 Distance:   sqrt ( (7 - -1)^2 + (7-1)^2 ) = sqrt( 8^2 + 6^2) = sqrt ( 64 + 36) = sqrt(100) = 10 

 

   Slope:   (7-1)/(7 - -1) = 6/8 = 3/4

 

YES, it works!!!

 

 THe other endpoint is (7,7)