Quadrilateral abcd is located at a(−2, 2), b(−2, 4), c(2, 4), and d(2, 2). the quadrilateral is then transformed using the rule (x + 7, y − 1) to form the image a'b'c'd'. what are the new coordinates of a', b', c', and d'? describe what characteristics you would find if the corresponding vertices were connected with line segments.

For each point, add 7 to the x coordinate and take away 1 from the y-coordinate. 

I am using a "tick-mark" to show the corresponding point, after "translation" (movement in a straight line). 

A(-2, 2) becomes A'(5, 1) 

Because all points have been translated in a straight line, in the same direction (the direction [7 -1]) by the same distance, the new figure A'B'C'D' will have the same shape, same size and same orientation as the original image.  

It will simply be in a different location on the graph. 

If you were to join A with A', B with B', and so on, you would see that all 4 vectors are parallel and are of equal length. 

If you were to compare the segment AB (side AB of the original quadrilateral) with the segment A'B' (in the new quadrilateral), you would see that they are parallel (same slope) and have the same length. 
Same with the other sides. 

Tow figures that are identical (except for their location) are said to be "congruent". 

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The new rectangle will be far to the right (7 units in x) and a bit below (-1 in y) from the old one. 

You should draw the two figures on a graph, and you will see a lot of this very clearly.