To solve the problem of determining which condition must hold true for a parallelogram [tex]\( ABCD \)[/tex] to be a rectangle, we should consider the geometric properties and algebraic conditions that characterize rectangles within a parallelogram setting.
For a parallelogram to be a rectangle, two key properties must be met:
1. The opposite sides must be parallel and equal in length, which is a defining feature of all parallelograms.
2. The adjacent sides must be perpendicular to each other, which distinguishes rectangles from other types of parallelograms.
Perpendicularity can be verified by checking that the slopes of adjacent sides are negative reciprocals of each other.
The slopes of sides [tex]\(AB\)[/tex], [tex]\(BC\)[/tex], [tex]\(CD\)[/tex], and [tex]\(DA\)[/tex] can be calculated using the coordinates of vertices:
- Slope of [tex]\(AB\)[/tex] ([tex]\(m_1\)[/tex]) is given by [tex]\( \frac{y_2 - y_1}{x_2 - x_1} \)[/tex]
- Slope of [tex]\(BC\)[/tex] ([tex]\(m_2\)[/tex]) is given by [tex]\( \frac{y_3 - y_2}{x_3 - x_2} \)[/tex]
- Slope of [tex]\(CD\)[/tex] ([tex]\(m_3\)[/tex]) is given by [tex]\( \frac{y_4 - y_3}{x_4 - x_3} \)[/tex]
- Slope of [tex]\(DA\)[/tex] ([tex]\(m_4\)[/tex]) is given by [tex]\( \frac{y_1 - y_4}{x_1 - x_4} \)[/tex]
For the parallelogram [tex]\(ABCD\)[/tex] to become a rectangle, the slopes of two consecutive sides must be negative reciprocals of each other:
- [tex]\( m_1 \times m_2 = -1 \)[/tex]
- [tex]\( m_2 \times m_3 = -1 \)[/tex]
- [tex]\( m_3 \times m_4 = -1 \)[/tex]
- [tex]\( m_4 \times m_1 = -1 \)[/tex]
Given this information, we now analyze the given choices:
A. [tex]\(\left(\frac{m_4-n_1}{z_4-z_3} = \frac{n-n_2}{z_1-z_2}\right)\)[/tex] and [tex]\(\left(\frac{z_4-n_1}{z_4-z_3} \times \frac{n-n_2}{z_3-z_3}\right) = -1\)[/tex]
This condition talks about ratios and perpendicularity but doesn't clearly relate to adjacent sides of the parallelogram [tex]\(ABCD\)[/tex].
C. [tex]\(\left(\frac{m_1-n_1}{z_4-x_3} = \frac{n_n-n_1}{z_1-z_1}\right)\)[/tex] and [tex]\(\left(\frac{m_1-n_2}{z_1-z_3} \times \frac{n_n-n_2}{z_3-x_2}\right) = -1\)[/tex]
Similar to choice A, it involves certain ratios and perpendicular conditions, but the indices and variables used are confusing and don't represent the consecutive sides correctly in the context of our problem.
D. [tex]\(\left(\frac{z_4-n_1}{z_1-z_3} = \frac{n_1-n_2}{z_3-z_1}\right)\)[/tex] and [tex]\(\left(\frac{y_4-n_1}{z_1-z_3} \times \frac{z_2-n_2}{z_2-z_1}\right) = -1\)[/tex]
This condition more directly relates to the slopes of the sides of the parallelogram, specifically checking if the product of the slopes of two adjacent sides equals [tex]\(-1\)[/tex]. This indicates a perpendicular relationship necessary for [tex]\(ABCD\)[/tex] to be a rectangle.
From this analysis, we select option D as the condition that must be true for parallelogram [tex]\(ABCD\)[/tex] to be a rectangle.
Therefore, the correct answer is:
3
