Chapter 5.4, Problem 21WE

To determine

To plot: the coordinates of three vertices of rectangle and to find the coordinates of the fourth vertex.

Also to check if the resulting figure is a rectangle or a square.

Answer to Problem 21WE

The coordinates of the fourth vertex is $D(2,5)$

The resulting figure is a rectangle

Explanation of Solution

Given Information:The coordinates of three verticesof a rectangle $A(2,1),B(4,1),C(4,5),D(\stackrel{?}{\mathrm{.....}},\stackrel{?}{\mathrm{......}})$ .

Graph: The coordinates of the three vertices are as shown in the graph below.Since the three given points are part of a rectangle, the fourth point must have coordinates in a way to complete the rectangle. The coordinates of the last point are also shown in the graph.

  

Interpretation: It has been specified that, the three given vertices are that of a rectangle and it is been plotted as in the above figure. So, based on the property of rectangle, every angle should be a right angle and this brings in a restriction on the location of the fourth vertex.

As visible from the graph, the fourth vertex is the intersection point of the perpendiculars from both points D and B and it has to be $D(2,5)$ . The resulting quadrilateral is a rectangle itself. It cannot be a square since the sides are not equal.