Question

if a(x1, y1), b(x2, y2), c(x3, y3), and d(x4, y4) form two line segments, $overline{ab}$ and $overline{cd}$, which condition needs to be met to prove $overline{ab} perp overline{cd}$?
a. $\frac{y_{4}-y_{3}}{x_{4}-x_{3}} \times \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = 1$
b. $\frac{y_{4}-y_{3}}{y_{2}-x_{1}}+\frac{x_{4}-x_{3}}{x_{2}-x_{1}} = 0$
c. $\frac{y_{4}-y_{3}}{x_{4}-x_{3}} \times \frac{y_{2}-y_{1}}{x_{2}-x_{1}}=-1$
d. $\frac{y_{2}-y_{1}}{x_{4}-x_{3}}-\frac{x_{2}-x_{1}}{y_{4}-y_{3}} = 1$
e. $\frac{y_{4}-y_{3}}{y_{2}-x_{1}}+\frac{x_{4}-x_{3}}{x_{2}-x_{1}} = 0$
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Explanation:

Step1: Recall slope - formula

The slope of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. The slope of line segment $\overline{AB}$ with $A(x_1,y_1)$ and $B(x_2,y_2)$ is $m_{AB}=\frac{y_2 - y_1}{x_2 - x_1}$, and the slope of line segment $\overline{CD}$ with $C(x_3,y_3)$ and $D(x_4,y_4)$ is $m_{CD}=\frac{y_4 - y_3}{x_4 - x_3}$.

Step2: Recall perpendicular - line condition

Two non - vertical lines with slopes $m_1$ and $m_2$ are perpendicular if and only if $m_1\times m_2=- 1$. So for $\overline{AB}\perp\overline{CD}$, we need $m_{AB}\times m_{CD}=-1$, which means $\frac{y_4 - y_3}{x_4 - x_3}\times\frac{y_2 - y_1}{x_2 - x_1}=-1$.

Answer:

C. $\frac{y_4 - y_3}{x_4 - x_3}\times\frac{y_2 - y_1}{x_2 - x_1}=-1$