Chapter 4.4, Problem 3CYP

To determine

The coordinates of the vertices of the rectangle $W\text{'}X\text{'}Y\text{'}Z\text{'}$ so that the perimeter of the rectangle $WXYZ$ image is one fourth of the preimage.

Explanation of Solution

Given Information:

The coordinates of the vertices of the rectangle WXYZ is

  $\begin{array}{l}W(4,4)\\ X(4,12)\\ Y(8,4)\\ Z(8,12)\end{array}$

Calculation:

Now the vertex matrix of the rectangle WXYZ is,

  $\left|\begin{array}{cccc}4& 4& 8& 8\\ 4& 12& 4& 12\end{array}\right|$

If the perimeter of the image is one fourth that of the preimage, then the lengths of the sides of the figures will be one fourth of the original length. So multiplying the vertex matrix with one fourth, we get the coordinates of the image triangle.

Now the vertex matrix of the triangle $W\text{'}X\text{'}Y\text{'}Z\text{'}$ is,

  $\left|\begin{array}{cccc}4& 4& 8& 8\\ 4& 12& 4& 12\end{array}\right|\times \frac{1}{4}=\left|\begin{array}{cccc}1& 1& 2& 2\\ 1& 3& 1& 3\end{array}\right|$

Thus, the coordinates of the vertices of the rectangle $W\text{'}X\text{'}Y\text{'}Z\text{'}$ is

  $\begin{array}{l}W\text{'}(1,1)\\ X\text{'}(1,3)\\ Y\text{'}(2,1)\\ Z\text{'}(2,3)\end{array}$