Question

randolph is creating rectangle wxyz so that wx has an equation of $y = \frac{1}{4}x + 4$. segment xy must pass through the point $(-2, 6)$. which of the following is the equation for xy?

• $y - 6 = \frac{1}{4}(x - (-2))$
• $y - (-2) = \frac{1}{4}(x - 6)$
• $y - 6 = -4(x - (-2))$
• $y - (-2) = -4(x - 6)$

Explanation:

Step1: Determine the slope of \(XY\)

Since \(WXYZ\) is a rectangle, \(WX\perp XY\). The slope of \(WX\) is \(m_{WX}=\frac{1}{4}\). For two perpendicular lines with slopes \(m_1\) and \(m_2\), \(m_1\times m_2=- 1\). Let \(m_{XY}\) be the slope of \(XY\). Then \(\frac{1}{4}\times m_{XY}=-1\), so \(m_{XY}=-4\).

Step2: Use the point - slope form

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)\) is a point on the line and \(m\) is the slope. The line \(XY\) passes through the point \((x_1=-2,y_1 = 6)\) and has a slope \(m=-4\). Substituting these values into the point - slope form gives \(y - 6=-4(x-(-2))\).

Answer:

\(y - 6=-4(x-(-2))\) (the third option)