Solve the Problem $\mathrm { A } ( - 1 , - 2 ) , \mathrm { B } ( 2 , - 3 )$

Solve the problem.
-Points $\mathrm { A } ( - 1 , - 2 ) , \mathrm { B } ( 2 , - 3 )$ , and $\mathrm { C } ( - 3 , - 8 )$ are given. Find all possible coordinates of point $\mathrm { D }$ such that the four points form a rectangle. For each location of point D found, find a translation that will take $\overline { \mathrm { AB } }$ to the opposite side of the rectangle formed.

A) $\mathrm { D } ( 1 , - 11 )$ ; A translation that takes $\overline { \mathrm { AB } }$ to $\overline { \mathrm { CD } }$ such that the image of $\mathrm { A }$ is $\mathrm { C }$ and the image of $\mathrm { B }$ is $D$ is $( x , y ) \rightarrow ( x + 1 , y + 3 )$ .
B) $\mathrm { D } ( 3 , - 9 )$ ; $A$ translation that takes $\overline { \mathrm { AB } }$ to $\overline { \mathrm { CD } }$ such that the image of $\mathrm { A }$ is $\mathrm { C }$ and the image of $\mathrm { B }$ is $D$ is $( x , y ) \rightarrow ( x - 3 , y - 1 )$ .
C) $\mathrm { D } ( 0 , - 9 )$ ; A translation that takes $\overline { \mathrm { AB } }$ to $\overline { \mathrm { CD } }$ such that the image of $\mathrm { A }$ is $\mathrm { C }$ and the image of $\mathrm { B }$ is $D$ is $( x , y ) \rightarrow ( x - 2 , y - 6 )$ .
D) $\mathrm { D } ( - 2 , - 8 )$ ; A translation that takes $\overline { \mathrm { AB } }$ to $\overline { \mathrm { CD } }$ such that the image of $\mathrm { A }$ is $\mathrm { C }$ and the image of $\mathrm { B }$ is $D$ is $( x , y ) \rightarrow ( x - 6 , y - 2 )$ .

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