Question 1117937: Find a point D(x,y) such that the points A(-3,1), B(4,0), C(0,-3) and D are the corners of a square. Justify your answer.
Found 2 solutions by solver91311, greenestamps:
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Three points define three line segments. Since only two of the segments could possibly be the sides of a square, we need to find which two, if any, are mutually perpendicular. Note that we cannot have two parallel segments because we only have three points.
Using the slope formula,  , segments  and  are found to be perpendicular by virtue of the fact that their slopes are mutually perpendicular.. You will have to do this arithmetic for yourself both to be able to show your work and because you will need the slope numbers later in this problem.
Now that we know we at least have a rectangle, it is necessary to verify that the two perpendicular segments have equal measures using the distance formula: ^2\ +\ (y_1\ -\ y_2)^2}) . Passing this test ensures that a square, as described in the problem statement is possible.
Then next thing needed is an equation of a line parallel to segment through point  , and the equation of a line parallel to segment through the point  . These equations are derived by using the Point-Slope form of an equation of a line and applying the fact that parallel lines have identical slopes. ) . The two slope numbers were developed in part one of this problem.
Once the two equations have been derived, Point D is the intersection of the two lines. Taking the two equations as a 2X2 simultaneous system, solve for the ordered pair that represents the intersection. Note that you are guaranteed to have a solution because the slopes of the two lines are not equal.
John
My calculator said it, I believe it, that settles it
Answer by greenestamps(13195)  (Show Source):
You can put this solution on YOUR website!
The solution by the other tutor is valid; but it is FAR more work than is necessary....
(1) Sketch a coordinate system with the three given points plotted.
(2) Connect the three points with line segments to form a triangle.
(3) It should be obvious which two of the three line segments are perpendicular and can therefore be sides of a square.
(4) Observe that going from A to C you move 3 units right and 4 units down; and going from C to B you go 4 units right and 3 units up. That confirms that segments AC and CB are both congruent and perpendicular.
(5) The 4th corner of the square will be found by going either 3 units left and 4 units up from B, or by going 4 units right and 3 units up from A. Either way gets you to the answer, (1,4).
Here is a diagram that I hope shows the process for finding the solution to the problem. Basically, you are solving the problem by drawing a bunch of congruent right triangles in the appropriate orientations.
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