Chapter 4.6, Problem 28SR

To determine

Whether the quadrilateral formed by equations $y=-2x,2x+y=6,y=\frac{1}{2}x+6\text{ and }x-2y=9$ represents a rectangle.

Answer to Problem 28SR

Yes, the quadrilateral formed by $y=-2x,2x+y=6,y=\frac{1}{2}x+6\text{ and }x-2y=9$ represents a rectangle.

Explanation of Solution

Given: The given quadrilateral has the equations,

  $\begin{array}{c}y=-2x\mathrm{......}\left(1\right)\\ 2x+y=\mathrm{6......}\left(2\right)\\ y=\frac{1}{2}x+\mathrm{6......}\left(3\right)\\ x-2y=\mathrm{9......}\left(4\right)\end{array}$

Calculation: With the help of given data of equations, the slopes for (1), (2), (3) and (4) are written as,

  $\begin{array}{c}\text{Slope of}\left(1\right)=-2\\ \text{Slope of}\left(2\right)=-2\\ \text{Slope of}\left(3\right)=\frac{1}{2}\\ \text{Slope of}\left(4\right)=\frac{1}{2}\end{array}$

Since, the lines (1) and (2) have same slopes so they are parallel. Similarly lines (3) and (4) have also same slopes. So they are also parallel. Now, slopes of (1) and (3), (2) and (4) are negative reciprocal to each other. Hence, the lines (1), (2) are perpendicular to (3) and (4). So the given lines forms a rectangle.