To find the equation of segment XY in rectangle WXYZ, let's start by analyzing the given information.
1. Equation of WX: It is provided as [tex]\( y = \frac{1}{4}x + 4 \)[/tex].
- This gives us the slope of WX, which is [tex]\(\frac{1}{4}\)[/tex].
2. Point XY Must Pass Through: The point [tex]\((-2, 6)\)[/tex].
3. Finding the Slope of XY:
- Since WX and XY are parts of a rectangle, XY must be perpendicular to WX.
- The slopes of perpendicular lines are negative reciprocals of each other.
- Therefore, the slope of XY will be the negative reciprocal of [tex]\(\frac{1}{4}\)[/tex].
- The negative reciprocal of [tex]\(\frac{1}{4}\)[/tex] is [tex]\(-4\)[/tex].
4. Equation of Line XY:
- We use the point-slope form of the equation of a line: [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Here, [tex]\((x_1, y_1)\)[/tex] is [tex]\((-2, 6)\)[/tex] and the slope [tex]\(m\)[/tex] is [tex]\(-4\)[/tex].
5. Substitute the Values:
- Plugging into the point-slope form:
[tex]\[ y - 6 = -4(x - (-2)) \][/tex]
Therefore, the equation for the segment XY is:
[tex]\[ \boxed{y - 6 = -4(x - (-2))} \][/tex]
So, the correct option is:
[tex]\[ \boxed{y - 6 = -4(x - (-2))} \][/tex]
