Given:
A(x1, y1), B(x2, y2), C(x3, y3), D(x4, y4).
Where AB and CD form two line segements.
Let's determine the condition which shows that AB is perpendicular to CD.
The slope of a perpendicular line is the negative reciprocal of the slope of the other line.
To show two lines are perpdincular, apply the formula:
[tex]m_{AB}\times m_{CD}=-1[/tex]
Where m is the slope.
Now, apply the slope formula:
[tex]\begin{gathered} m_{AB}=\frac{y2-y1}{x2-x1} \\ \\ m_{CD}=\frac{y4-y3}{x4-x3} \end{gathered}[/tex]
Thus, we have:
[tex]\frac{y4-y3}{x4-x3}\times\frac{y2-y1}{x2-x1}=-1[/tex]
Therefore, the condition that needs to be met to prove that AB is perpendicular to CD is:
[tex]\frac{y4-y3}{x4-x3}\times\frac{y2-y1}{x2-x1}=-1[/tex]
ANSWER: C
[tex]\frac{y4-y3}{x4-x3}\times\frac{y2-y1}{x2-x1}=-1[/tex]
