Question

1. overline{ab} of rectangle abcd passes through the point (2, 0) and is perpendicular to the graph of ( y = \frac{1}{4}x - 3 ). ( overline{cd} ) is parallel to ( overline{ab} ) and passes through point (-1, 6). select the equation in slope-intercept form of the line that includes ( overline{cd} ).

a ( y = \frac{1}{4}x + 2 )
b ( y = -\frac{1}{4}x + 2 )
c ( y = -4x + 2 )
d ( y = 4x + 2 )

Explanation:

Step1: Find slope of AB

The line AB is \( y = \frac{1}{4}x - 3 \). The slope of AB (\( m_{AB} \)) is \( \frac{1}{4} \) (from \( y = mx + b \) form).

Step2: Determine slope of CD

Since CD is perpendicular to AB, the slope of CD (\( m_{CD} \)) is the negative reciprocal of \( \frac{1}{4} \). So \( m_{CD} = -4 \) (because \( \text{negative reciprocal of } \frac{a}{b} \text{ is } -\frac{b}{a} \), here \( a = 1, b = 4 \), so \( -\frac{4}{1} = -4 \)).

Step3: Use point - slope form for CD

We know CD passes through \( (-1, 6) \) and has slope \( -4 \). The point - slope form is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1)=(-1, 6) \) and \( m=-4 \).
Substitute values: \( y - 6 = -4(x - (-1)) \) which simplifies to \( y - 6 = -4(x + 1) \).

Step4: Convert to slope - intercept form

Expand the right - hand side: \( y - 6 = -4x - 4 \).
Add 6 to both sides: \( y = -4x - 4 + 6 \), so \( y = -4x + 2 \).

Answer:

C. \( y = -4x + 2 \)