To determine if the two line segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are perpendicular, we need to use the concept of slopes. The slopes of two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex].
1. Calculate the slope of [tex]\(\overline{AB}\)[/tex]:
The slope of a line segment [tex]\(\overline{AB}\)[/tex] connecting points [tex]\(A(x_1, y_1)\)[/tex] and [tex]\(B(x_2, y_2)\)[/tex] is given by:
[tex]\[
\text{slope}_{AB} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
2. Calculate the slope of [tex]\(\overline{CD}\)[/tex]:
The slope of a line segment [tex]\(\overline{CD}\)[/tex] connecting points [tex]\(C(x_3, y_3)\)[/tex] and [tex]\(D(x_4, y_4)\)[/tex] is given by:
[tex]\[
\text{slope}_{CD} = \frac{y_4 - y_3}{x_4 - x_3}
\][/tex]
3. Check if the product of the slopes is [tex]\(-1\)[/tex]:
For the two line segments to be perpendicular, their slopes must satisfy:
[tex]\[
\text{slope}_{AB} \times \text{slope}_{CD} = -1
\][/tex]
Substituting the respective slopes, we get:
[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]
4. Simplify the condition:
Simplifying the above condition, it can be written as:
[tex]\[
\frac{(y_2 - y_1) \times (y_4 - y_3)}{(x_2 - x_1) \times (x_4 - x_3)} = -1
\][/tex]
Rearranging, we obtain:
[tex]\[
(y_2 - y_1) \times (y_4 - y_3) = - (x_2 - x_1) \times (x_4 - x_3)
\][/tex]
Among the given options, the correct condition that matches this analysis is option [tex]\( \text{C}\)[/tex]:
[tex]\[
\frac{y_2 - y_1}{x_2 - x_1} \times \frac{y_4 - y_3}{x_4 - x_3} = -1
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{3}
\][/tex]
