To determine whether a given parallelogram is a rectangle, we need to ensure that it has right angles. This implies that the slopes of the adjacent sides must be negative reciprocals of each other.
Let's examine the vertices of the parallelogram [tex]\( A(x_1, y_1) \)[/tex], [tex]\( B(x_2, y_2) \)[/tex], [tex]\( C(x_3, y_3) \)[/tex], and [tex]\( D(x_4, y_4) \)[/tex].
For these coordinates to form a rectangle, the slopes of the sides need to satisfy certain conditions:
1. The slopes of opposite sides must be equal because that's a property of parallelograms and rectangles (i.e., parallel sides).
2. The product of the slopes of adjacent sides must be [tex]\(-1\)[/tex] since the sides are perpendicular to each other in a rectangle.
Let's consider the slopes of the sides:
- Slope of [tex]\( AB \)[/tex] or [tex]\( CD \)[/tex]:
[tex]\[
\text{slope}_{AB} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
- Slope of [tex]\( BC \)[/tex] or [tex]\( AD \)[/tex]:
[tex]\[
\text{slope}_{BC} = \frac{y_3 - y_2}{x_3 - x_2}
\][/tex]
- Slope of [tex]\( CD \)[/tex] or [tex]\( AB \)[/tex]:
[tex]\[
\text{slope}_{CD} = \frac{y_4 - y_3}{x_4 - x_3}
\][/tex]
- Slope of [tex]\( DA \)[/tex] or [tex]\( BC \)[/tex]:
[tex]\[
\text{slope}_{DA} = \frac{y_1 - y_4}{x_1 - x_4}
\][/tex]
Now, for the parallelogram to be a rectangle:
1. [tex]\(\text{slope}_{AB} = \text{slope}_{CD}\)[/tex]
2. [tex]\(\text{slope}_{BC} = \text{slope}_{DA}\)[/tex]
3. [tex]\(\text{slope}_{AB} \times \text{slope}_{BC} = -1 \)[/tex]
4. [tex]\(\text{slope}_{DA} \times \text{slope}_{BC} = -1 \)[/tex]
Considering the given options:
- Option [tex]\(B\)[/tex] is:
[tex]\[
\left(\frac{y_4 - y_3}{x_4 - x_3} = \frac{y_2 - y_1}{x_2 - x_1}\right) \quad \text{and} \quad \left(\frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_2 - y_1}{x_2 - x_1} = -1\right)
\][/tex]
This matches our criteria:
- The first part ensures that opposite sides [tex]\( AB \)[/tex] and [tex]\( CD \)[/tex] have the same slope.
- The second part ensures that the product of the slopes of adjacent sides is [tex]\(-1\)[/tex], thus ensuring right angles between them.
Therefore, option [tex]\( B \)[/tex] is the correct answer.
