To determine whether the line segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are perpendicular, we need to use the concept of the slopes of these line segments. The property that indicates perpendicularity is that the product of the slopes of two perpendicular lines is [tex]\(-1\)[/tex].
Let's denote the coordinates of the points as follows:
- [tex]\(A(x_1, y_1)\)[/tex]
- [tex]\(B(x_2, y_2)\)[/tex]
- [tex]\(C(x_3, y_3)\)[/tex]
- [tex]\(D(x_4, y_4)\)[/tex]
We will calculate the slopes of the line segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex].
1. Slope of [tex]\(\overline{AB}\)[/tex]:
The slope [tex]\(m_1\)[/tex] of the line segment [tex]\(\overline{AB}\)[/tex] can be calculated using the formula:
[tex]\[
m_1 = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
provided that [tex]\(x_2 \neq x_1\)[/tex].
2. Slope of [tex]\(\overline{CD}\)[/tex]:
The slope [tex]\(m_2\)[/tex] of the line segment [tex]\(\overline{CD}\)[/tex] can be calculated using the formula:
[tex]\[
m_2 = \frac{y_4 - y_3}{x_4 - x_3}
\][/tex]
provided that [tex]\(x_4 \neq x_3\)[/tex].
For the line segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] to be perpendicular, the product of their slopes must be [tex]\(-1\)[/tex]:
[tex]\[
m_1 \times m_2 = -1
\][/tex]
Substituting the formulas for the slopes, the condition becomes:
[tex]\[
\left(\frac{y_2 - y_1}{x_2 - x_1}\right) \times \left(\frac{y_4 - y_3}{x_4 - x_3}\right) = -1
\][/tex]
Simplifying, we get:
[tex]\[
\frac{(y_2 - y_1) \times (y_4 - y_3)}{(x_2 - x_1) \times (x_4 - x_3)} = -1
\][/tex]
Hence, the correct condition to prove that [tex]\(\overline{AB} \perp \overline{CD}\)[/tex] is:
[tex]\[
\frac{(y_4 - y_3)}{(x_4 - x_3)} \times \frac{(y_2 - y_1)}{(x_2 - x_1)} = -1
\][/tex]
Comparing this with the given options, we see that option C matches:
[tex]\[
C. \frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_2 - y_1}{x_2 - x_1} = -1
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{C}
\][/tex]
