To determine if a parallelogram [tex]\(ABCD\)[/tex] with vertices [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] is actually a rectangle, there are specific conditions we need to check.
A parallelogram becomes a rectangle if all the angles are 90 degrees. For angles to be 90 degrees, the dot product of adjacent sides must equal zero. This is because the dot product of two vectors is zero if and only if the vectors are perpendicular, implying a right angle between them.
Given:
- Vector from [tex]\(A\)[/tex] to [tex]\(B\)[/tex]: [tex]\(\vec{AB} = (x_2 - x_1, y_2 - y_1)\)[/tex]
- Vector from [tex]\(A\)[/tex] to [tex]\(D\)[/tex]: [tex]\(\vec{AD} = (x_4 - x_3, y_4 - y_3)\)[/tex]
To check if [tex]\(\angle DAB\)[/tex] is 90 degrees, we utilize the dot product of [tex]\(\vec{AB}\)[/tex] and [tex]\(\vec{AD}\)[/tex]:
[tex]\[
\vec{AB} \cdot \vec{AD} = (x_2 - x_1)(x_4 - x_3) + (y_2 - y_1)(y_4 - y_3)
\][/tex]
This dot product should be zero for the angle [tex]\(\angle DAB\)[/tex] to be 90 degrees:
[tex]\[
(x_2 - x_1)(x_4 - x_3) + (y_2 - y_1)(y_4 - y_3) = 0
\][/tex]
Hence, statement A. [tex]\((x_2 - x_1)(x_4 - x_3) + (y_2 - y_1)(y_4 - y_3) = 0\)[/tex] must be true for parallelogram [tex]\(ABCD\)[/tex] to be a rectangle.
Additionally, another condition for a parallelogram to be a rectangle is that each of its internal angles must be 90 degrees. By the geometric properties, we understand that each internal angle in a rectangle is [tex]\(90\)[/tex] degrees.
Therefore, the answer should combine these two conditions:
1. [tex]\((x_2 - x_1)(x_4 - x_3) + (y_2 - y_1)(y_4 - y_3) = 0\)[/tex]
2. Each angle in a rectangle is [tex]\(90\)[/tex] degrees.
Thus, the correct statement ensuring [tex]\(ABCD\)[/tex] is a rectangle includes both these conditions:
A. [tex]\((x_2 - x_1)(x_4 - x_3) + (y_2 - y_1)(y_4 - y_3) = 0\)[/tex]
Each angle in a rectangle is 90 degrees.
The correct choice is option A.
