Question

Three vertices of a rectangle are (–3, 4), (5, 4), and (5, –2). 

What are the coordinates of the fourth vertex of the rectangle?  

         A. (–3, –4)                                                   

        B. (–3, –2)                                                   

      C. (5, –3)      

        D. (5, 2)

Answer 1

The rectangle doesn't necessarily have to be standing up straight or laying down flat
on the graph. It could be tilted ... leaning. Dealing with those coordinates would be
a nightmare, and since you're in Middle School and just learning this stuff, we can be
pretty sure that the rectangle in this problem is straight on the graph.

If we think about it for a second, and visualize a rectangle in our mind, we realize that:

-- Both top corners have the same y-coordinate.
-- Both bottom corners have the same y-coordinate.
-- Both corners on the left side have the same x-coordinate.
-- Both corners on the right side have the same x-coordinate.

The first two points given in the question are ( -3, 4) and (5, 4).

-- They have the same y-coordinate, so they must be either the top
or the bottom two corners.

-- We also know right away that the left side of the rectangle is at x = -3
and the right side of the rectangle is at x = 5 .

Now look at the third given point ... (5, -2) .

-- Its x-coordinate is 5 so we know it's a corner on the right side.

-- But we already had one corner on the right side ... the point at (5, 4).
This new one is lower.

-- So now we know that (5, 4) was the upper right corner, (-3, 4) was the
upper left corner, and this new point is the lower right corner.

All we're missing is the lower left corner.

We know now that the left side is at x = -3 and the right side is at 5 .
We also know now that the top is at y = 4 and the bottom is at y = -2 .

So the bottom left corner is (-3, -2). That's choice - 'B' .

Answer 2

`A =(-3, 4), \ \ B= (5, 4),\ \ C= (5, -2)\\ \\ The \ slope \ of \ AB : \\ \\ \\ m= (y_(2)-y_(1))/(x_(2)-x_(1) ) = (4-4)/(5+3)=(0)/(8)=0\\ \\ The \ slope \ of \ CB :\\ \\ m= (y_(2)-y_(1))/(x_(2)-x_(1) ) = (5-5)/(-2-4)=0`


`The \ slope \ of \ lines \ parallel \ is \ the \ same \\ \\To \ find \ the \ coordinates \ of \ the \ fourth \ vertex, \ add \ the \ coordinates \\ of \ A \ and \ C \ together, \ and \ subtract \ the \ coordinates \ of \ B. \\ \\ The \ fourth \ vertex \ is: (-3+5-5; 4-2-4) = (-3, -2) \\ \\ Answer: \ B. \ \ (-3, -2)`