To determine which ordered pairs could be points on a line that is perpendicular to the given line with a slope of [tex]\(-\frac{4}{5}\)[/tex], we need to follow several steps. A perpendicular line's slope is the negative reciprocal of the original line's slope.
Given:
- The slope of the original line is [tex]\( m = -\frac{4}{5} \)[/tex].
The slope of a line perpendicular to this one would be [tex]\( m_\perp = \frac{5}{4} \)[/tex]. We need to identify the sets of points from the options provided that have this slope between them.
### Steps:
1. Calculate the slope for each pair of points:
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
2. Check each pair of points:
- Option 1: [tex]\((-2, 0)\)[/tex] and [tex]\((2, 5)\)[/tex]
[tex]\[
m = \frac{5 - 0}{2 - (-2)} = \frac{5}{4}
\][/tex]
This pair has the slope [tex]\(\frac{5}{4}\)[/tex].
- Option 2: [tex]\((-4, 5)\)[/tex] and [tex]\((4, -5)\)[/tex]
[tex]\[
m = \frac{-5 - 5}{4 - (-4)} = \frac{-10}{8} = -\frac{5}{4}
\][/tex]
This pair does not have the slope [tex]\(\frac{5}{4}\)[/tex].
- Option 3: [tex]\((-3, 4)\)[/tex] and [tex]\((2, 0)\)[/tex]
[tex]\[
m = \frac{0 - 4}{2 - (-3)} = \frac{-4}{5}
\][/tex]
This pair does not have the slope [tex]\(\frac{5}{4}\)[/tex].
- Option 4: [tex]\((1, -1)\)[/tex] and [tex]\((6, -5)\)[/tex]
[tex]\[
m = \frac{-5 - (-1)}{6 - 1} = \frac{-4}{5}
\][/tex]
This pair does not have the slope [tex]\(\frac{5}{4}\)[/tex].
- Option 5: [tex]\((2, -1)\)[/tex] and [tex]\((10, 9)\)[/tex]
[tex]\[
m = \frac{9 - (-1)}{10 - 2} = \frac{10}{8} = \frac{5}{4}
\][/tex]
This pair has the slope [tex]\(\frac{5}{4}\)[/tex].
### Conclusion:
The pairs that form a line with a slope of [tex]\(\frac{5}{4}\)[/tex], and hence are perpendicular to the original line with slope [tex]\(-\frac{4}{5}\)[/tex], are:
- [tex]\((-2, 0)\)[/tex] and [tex]\((2, 5)\)[/tex]
- [tex]\((2, -1)\)[/tex] and [tex]\((10, 9)\)[/tex]
Thus, the ordered pairs that could be points on a line that is perpendicular to the given line are:
- [tex]\((-2, 0)\)[/tex] and [tex]\((2, 5)\)[/tex]
- [tex]\((2, -1)\)[/tex] and [tex]\((10, 9)\)[/tex]
