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Three vertices of a rectangle are (2, 5), (-3, 5), and (2, -1). Find the coordinates of the fourth vertex.
Solution:
The first thing that we need to do is to plot the three points as follows
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| Photo by Math Principles in Everyday Life |
Method 1:
Did you notice that the opposite sides of a rectangle are parallel to x-axis and y-axis? AD is parallel to x-axis and BC and CD is parallel to y-axis and AB. Points A and D have the same value of y which is 5. If AD is parallel to BC, then the value of y at point C must be -1 also.
Points A and B have the same value of x which is 2. If AB is parallel to CD, then the value of x at point C must be -3 also. Therefore, the fourth vertex of a rectangle is C (-3, -1).
Method 2:
This method is easier because you will need to use the midpoint formula only. We know that the center of a rectangle is the intersection of its diagonals. Their diagonals are equal and bisect each other at its center.
In this case, we can get the midpoint of a rectangle at point O by using the midpoint formula for BD as follows
Hence, the coordinates of the center of a rectangle is O (-½, 2).
If the midpoint of AC is also point O, then we can solve for the coordinates of point C by using the midpoint formula of AC as follows
Therefore, the fourth vertex of a rectangle is C (-3, -1).