Chapter 4.3, Problem 43PPS

a.

To determine

To graph the given points on coordinate plane.

Explanation of Solution

Given:

  $A(-3,3)$

  $B(3,5)$

  $C(-4,0)$

Graph:

Plot the given points in the graph:

x	y
-3	3
3	5
-4	0

  

Interpretation:

The given points are marked in the graph as above.

b.

To determine

To find the fourth coordinate of a parallelogram

Answer to Problem 43PPS

  $(-10,-2)$

Explanation of Solution

Given:

  $A(-3,3)$

  $B(3,5)$

  $C(-4,0)$

Formula used:

The diagonals of a parallelogram bisect each other.

Midpoint of line segments joining $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

Calculation:

Let the unknown point be D $(x,y)$

  $(x_1,y_1)=(-3,3)$

  $(x_2,y_2)=(3,5)$

  $(x_3,y_3)=(-4,0)$

In a parallelogram, the diagonals bisect each other.

  

Midpoint of BD = Midpoint of AC

Substitute the corresponding values of A $(x_1,y_1)$ and C $(x_3,y_3)$

Midpoint of line segments joining $(-3,3)$ and $(-4,0)$ is

  $\begin{array}{c}(\frac{x_1+x_2}{2},\frac{y_1+y_3}{2})=(\frac{-3-4}{2},\frac{3+0}{2})\\ =(\frac{-7}{2},\frac{3}{2})\end{array}$

Therefore, the midpoint of the parallelogram is $(\frac{-7}{2},\frac{3}{2})$ .

Midpoint of line segments joining $(3,5)$ and $(x,y)$ is

  $\begin{array}{c}(\frac{x_1+x_2}{2},\frac{y_1+y_3}{2})=(\frac{3+x}{2},\frac{5+y}{2})\\ (\frac{-7}{2},\frac{3}{2})=(\frac{3+x}{2},\frac{5+y}{2})\end{array}$

Equating the values of x,

  $\begin{array}{l}\Rightarrow \frac{3+x}{2}=\frac{-7}{2}\\ \Rightarrow 3+x=-7\\ \Rightarrow x=-10\end{array}$

Equating the values of y,

  $\begin{array}{l}\Rightarrow \frac{5+y}{2}=\frac{3}{2}\\ \Rightarrow 5+y=3\\ \Rightarrow y=-2\end{array}$

Therefore, the Fourth point is D $(-10,-2)$ .

Conclusion:

Therefore, the fourth point is $(-10,-2)$

c.

To determine

To find the minimum number of points to be moved so that the parallelogram becomes a rectangle.

Answer to Problem 43PPS

2

Explanation of Solution

Given:

Minimum number of points has to be moves to make the parallelogram into a rectangle.

Concept used:

When the parallelogram’s angles are right angles, it becomes a rectangle.

When any two points of a parallelogram are moved to make right angles, it becomes a parallelogram.

Here, when two points are moved, any two of the four points, the parallelogram becomes a rectangle.

Conclusion:

Therefore, the minimum of two points can be moved to make a parallelogram into a rectangle.