Question

The coordinates of parallelogram UVWZ are U(a, 0), W(c - a, b), and Z(c, 0). Find the coordinates of V without using any new variables.
1. (c,b)
2. (a,b)
3. (0,b)
4. (b,0)

Answer 1

Check the picture below.

so, as you can see, the UV segment is parallel to ZW, and therefore, they're the same slope, hmmm wait just a second, what is the slope of ZW anyway?

$\bf \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ % (a,b) &Z&(~ c &,& 0~) % (c,d) &W&(~ c-a &,& b~) \end{array} \\\\\\ % slope = m slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{b-0}{(c-a)-c}\implies \cfrac{b}{-a}$

since now we know the ZW slope, we also know what is the slope for UV, thus,

$\bf \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ % (a,b) &U&(~ a &,& 0~) % (c,d) &V&(~ x &,& y~) \end{array} \\\\\\ % slope = m slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{y-0}{x-a}~~=~~\stackrel{ZW's~slope}{\cfrac{b}{-a}} \\\\\\ \begin{cases} y-0=b\implies \boxed{y=b}\\ -----------\\ x-a=-a\implies \boxed{x=0} \end{cases}\qquad \qquad V~(0,b)$