To determine the correct equation of a line perpendicular to the line passing through points [tex]\( B \)[/tex] and [tex]\( D \)[/tex] for the given square [tex]\( ABCD \)[/tex] with center [tex]\( Z \)[/tex] at the origin, and vertices [tex]\( A(-20, -20) \)[/tex] and [tex]\( C(20, 20) \)[/tex], follow these steps:
1. Determine the Coordinates of Points [tex]\( B \)[/tex] and [tex]\( D \)[/tex]:
Given [tex]\( A(-20, -20) \)[/tex] and [tex]\( C(20, 20) \)[/tex], we can find [tex]\( B \)[/tex] and [tex]\( D \)[/tex] by using the symmetry of the square and keeping in mind that the center [tex]\( Z \)[/tex] is at the origin.
- The coordinates of [tex]\( B \)[/tex] can be derived by swapping the coordinates and symmetry around the origin, giving us [tex]\( B(-20, 20) \)[/tex].
- Similarly, the coordinates of [tex]\( D \)[/tex] are [tex]\( (20, -20) \)[/tex].
2. Find the Slope of Line [tex]\( BD \)[/tex]:
The slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
\text{slope}_{BD} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the coordinates of [tex]\( B(-20, 20) \)[/tex] and [tex]\( D(20, -20) \)[/tex]:
[tex]\[
\text{slope}_{BD} = \frac{-20 - 20}{20 - (-20)} = \frac{-40}{40} = -1
\][/tex]
3. Determine the Slope of the Perpendicular Line:
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Since the slope of line [tex]\( BD \)[/tex] is [tex]\(-1\)[/tex], the slope of the perpendicular line is:
[tex]\[
\text{slope}_{\text{perpendicular}} = -\left(\frac{1}{-1}\right) = 1
\][/tex]
4. Identify the Correct Equation:
From the given options, we need to find the equation that has a slope of [tex]\( 1 \)[/tex]:
- (A) [tex]\( y = -x \)[/tex] has a slope of [tex]\(-1\)[/tex].
- (B) [tex]\( y = x \)[/tex] has a slope of [tex]\( 1 \)[/tex].
- (C) [tex]\( y = -\frac{2}{3} x \)[/tex] has a slope of [tex]\(-\frac{2}{3} \)[/tex].
- (D) [tex]\( y = \frac{2}{3} x \)[/tex] has a slope of [tex]\(\frac{2}{3} \)[/tex].
Therefore, the equation of the line that is perpendicular to the line passing through points [tex]\( B \)[/tex] and [tex]\( D \)[/tex] and has a slope of [tex]\( 1 \)[/tex] is:
[tex]\[
\boxed{y = x}
\][/tex]
