Question

Segment AB is on the line y − 9 = −4(x + 1), and segment CD is on the line y − 6 = one fourth(x − 3). Which statement proves the relationship of segments AB and CD?

They are perpendicular because they have slopes that are opposite reciprocals of 4 and negative one fourth.
They are parallel because they have the same slope of 4.
They are perpendicular because they have slopes that are opposite reciprocals of −4 and one fourth.
They are parallel because they have the same slope of negative one fourth.

Answer 1

To determine the relationship between segments AB and CD, we need to find the slopes of the lines on which they lie.

1. For segment AB, the line is given by the equation:
\[
y - 9 = -4(x + 1)
\]
This can be rewritten in slope-intercept form (y = mx + b) as follows:
\[
y = -4x - 4 + 9 \\
y = -4x + 5
\]
Thus, the slope (m) of line AB is -4.

2. For segment CD, the line is given by the equation:
\[
y - 6 = \frac{1}{4}(x - 3)
\]
Similarly, rewriting it in slope-intercept form gives:
\[
y = \frac{1}{4}x - \frac{3}{4} + 6 \\
y = \frac{1}{4}x + \frac{21}{4}
\]
Hence, the slope (m) of line CD is \(\frac{1}{4}\).

Now we can analyze the relationship between the slopes of the two lines:
- The slope of AB is -4.
- The slope of CD is \(\frac{1}{4}\).

To establish whether the segments AB and CD are perpendicular, parallel, or neither, we note that two lines are perpendicular if the product of their slopes is -1.

Calculating the product of the slopes:
\[
(-4) \cdot \left(\frac{1}{4}\right) = -1
\]

Since their product is -1, we conclude that segments AB and CD are perpendicular.

Thus, the correct statement that proves the relationship of segments AB and CD is:
**They are perpendicular because they have slopes that are opposite reciprocals of -4 and one fourth.**