Chapter 4.4, Problem 43PPS

To determine

Plot the points $A\left(-3,3\right),B\left(3,5\right)\text{ and }C\left(-4,0\right)$ on coordinate plane.

Explanation of Solution

Given:

The given points are $A\left(-3,3\right),B\left(3,5\right)\text{ and }C\left(-4,0\right)$ .

Graph:The points $A\left(-3,3\right),B\left(3,5\right)\text{ and }C\left(-4,0\right)$ are plotted on coordinate plane as shown.

  

To determine

If the points $A\left(-3,3\right),B\left(3,5\right)\text{ and }C\left(-4,0\right)$ are three vertices of a parallelogram, then find the vertex $D$ .

Answer to Problem 43PPS

The point D is $\left(-10,-2\right)$.

Explanation of Solution

Given:The points $A\left(-3,3\right),B\left(3,5\right)\text{ and }C\left(-4,0\right)$ are three vertices of a parallelogram and considering that the fourth vertex is $D\left(a,b\right)$ .

Calculation:Since ABCD forms a parallelogram therefore AB is parallel to DC and BC is parallel to AD.

Now, by slope formula,

  $\begin{array}{c}{m}_{AB}=m_{DC}\\ \frac{5-3}{3+3}=\frac{0-b}{-4-a}\\ \frac{1}{3}=\frac{b}{a+4}\\ a+4=3b\\ a=3b-\mathrm{4......}\left(1\right)\end{array}$

Similarly,

  $\begin{array}{c}{m}_{BC}=m_{AD}\\ \frac{0-5}{-4-3}=\frac{b-3}{a+3}\\ \frac{5}{7}=\frac{b-3}{a+3}\\ 5a+15=7b-21\\ 5a=7b-\mathrm{36......}\left(2\right)\end{array}$

Solving (1) and (2), by plugging $a=3b-4$ in (2)

  $\begin{array}{c}5\left(3b-4\right)=7b-36\\ 15b-20=7b-36\\ 8b=-16\\ b=-2\end{array}$

Plugging $b=-2$ in (1),

  $\begin{array}{c}a=3\left(-2\right)-4\\ a=-10\\ \therefore D\text{is }\left(-10,-2\right)\mathrm{.......}\left(3\right)\end{array}$

Hence, the coordinate of point D is $\left(-10,-2\right)$.

To determine

The minimum number of points that could be moved to make the parallelogram a rectangle. Also, find the point that should be moved.

Answer to Problem 43PPS

The minimum number of point that should be moved are any two points.

Explanation of Solution

Given: The points $A\left(-3,3\right),B\left(3,5\right)\text{ and }C\left(-4,0\right)$ are three vertices of a parallelogram and considering that the fourth vertex is $D\left(a,b\right)$ .

Calculation: From (3), the fourth vertex is $D\left(-10,-2\right)$ . And the lines AB, DC have same slope. Similarly, lines BC, AD will have same slope for ABCD to be a parallelogram. Hence, if we move any two points D and C for our convenience so that the slopes of line AB and DC remains equal because of their parallel property, then move any two points D and C such that the slopes of line AD and BC become undefined. When the slopes of line AD and BC are undefined then then parallelogram becomes a rectangle.