Answered

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Randolph is creating rectangle WXYZ so that WX has an equation of [tex] y = \frac{1}{4} x + 4 [/tex]. Segment XY must pass through the point [tex] (-2, 6) [/tex]. Which of the following is the equation for XY?

A. [tex] y - 6 = \frac{1}{4}(x - (-2)) [/tex]
B. [tex] y - (-2) = \frac{1}{4}(x - 6) [/tex]
C. [tex] y - 6 = -4(x - (-2)) [/tex]
D. [tex] y - (-2) = -4(x - 6) [/tex]

Sagot :

GinnyAnswer
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]
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