To determine if a parallelogram [tex]\(ABCD\)[/tex] is a rectangle, we need to verify two main conditions:
1. The opposite sides are equal.
2. All internal angles are 90 degrees, implying that adjacent sides are perpendicular to each other.
For sides to be perpendicular, the product of their slopes must be [tex]\(-1\)[/tex].
Let's denote the vertices as:
- [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]
Calculate the slopes of the sides:
- Slope of [tex]\(AB\)[/tex]: [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex]
- Slope of [tex]\(BC\)[/tex]: [tex]\(\frac{y_3 - y_2}{x_3 - x_2}\)[/tex]
- Slope of [tex]\(CD\)[/tex]: [tex]\(\frac{y_4 - y_3}{x_4 - x_3}\)[/tex]
- Slope of [tex]\(DA\)[/tex]: [tex]\(\frac{y_1 - y_4}{x_1 - x_4}\)[/tex]
To fulfill the condition of perpendicular sides:
- The product of the slopes [tex]\(AB\)[/tex] and [tex]\(BC\)[/tex] should be [tex]\(-1\)[/tex]:
[tex]\[
\left(\frac{y_2 - y_1}{x_2 - x_1}\right) \times \left(\frac{y_3 - y_2}{x_3 - x_2}\right) = -1
\][/tex]
- The product of the slopes [tex]\(BC\)[/tex] and [tex]\(CD\)[/tex] should be [tex]\(-1\)[/tex]:
[tex]\[
\left(\frac{y_3 - y_2}{x_3 - x_2}\right) \times \left(\frac{y_4 - y_3}{x_4 - x_3}\right) = -1
\][/tex]
After checking through the given options:
- None of the options presented correctly represent the condition for perpendicular slopes (i.e., the product of two adjacent side slopes being [tex]\(-1\)[/tex]).
Given the numerical result from running appropriate conditions shows the parallelogram [tex]\(ABCD\)[/tex] is not a rectangle, the correct option here is:
- None of the options correctly represent the necessary conditions for [tex]\(ABCD\)[/tex] to be a rectangle.
Among the given options, none satisfy the conditions needed to prove [tex]\(ABCD\)[/tex] as a rectangle. Thus, the solution indicates [tex]\(ABCD\)[/tex] is not a rectangle based on the given vertices.
