Question

segment ab is on the line $y - 9 = -4(x + 1)$, and segment cd is on the line $y - 6 = \frac{1}{4}(x - 3)$. which statement proves the relationship of segments ab and cd?

1. they are perpendicular because they have slopes that are opposite reciprocals of 4 and $-\frac{1}{4}$
2. they are parallel because they have the same slope of 4
3. they are perpendicular because they have slopes that are opposite reciprocals of -4 and $\frac{1}{4}$
4. they are parallel because they have the same slope of $-\frac{1}{4}$

Explanation:

Step1: Identify slope of AB

The line for AB is in point-slope form \(y - y_1 = m(x - x_1)\), where \(m\) is the slope. For \(y - 9 = -4(x + 1)\), the slope \(m_1 = -4\).

Step2: Identify slope of CD

For the line of CD \(y - 6 = \frac{1}{4}(x - 3)\), the slope \(m_2 = \frac{1}{4}\).

Step3: Check perpendicular condition

Two lines are perpendicular if \(m_1 \times m_2 = -1\). Calculate: \(-4 \times \frac{1}{4} = -1\), so they are perpendicular, with slopes that are opposite reciprocals of \(-4\) and \(\frac{1}{4}\).

Answer:

They are perpendicular because they have slopes that are opposite reciprocals of -4 and $\frac{1}{4}$