Chapter 12.8, Problem 22E

To determine

To find: The midpoint of a side in a third quadrant.

Answer to Problem 22E

Themidpoint of the side in the third quadrant is $\left(\frac{-w}{2},\frac{-v}{2}\right)$ .

Explanation of Solution

Given information:

The vertices of a parallelogram $\left(w,0\right),\left(0,v\right),\left(-w,0\right),\left(0,-v\right)$ .

  

The coordinate of the midpoint between two points can be obtained by,

  $m=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Here,

it can be clearly seen from the diagram that the points in the third quadrant are,

   $x_1=-u,x_2=0,y_1=0,y_2=-v$

Substitute the values in the given formula,

  $\begin{array}{c}m=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\end{array}$

    $\begin{array}{c}=\left(\frac{-w+0}{2},\frac{0-v}{2}\right)\end{array}$

    $\begin{array}{c}=\left(\frac{-w}{2},\frac{-v}{2}\right)\end{array}$

Therefore, the midpoint of the side in the third quadrant is $\left(\frac{-w}{2},\frac{-v}{2}\right)$ .

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