## Friday, June 20, 2014

### Square, Rectangle, Parallelogram Problems, 10

Category: Analytic Geometry, Plane Geometry

"Published in Vacaville, California, USA"

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 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).