To determine which ordered pairs of points lie on a line that is perpendicular to a line with slope [tex]\(-\frac{4}{5}\)[/tex], we must find the negative reciprocal of the original line's slope. The negative reciprocal of [tex]\(-\frac{4}{5}\)[/tex] is [tex]\(\frac{5}{4}\)[/tex]. Therefore, the perpendicular line will have a slope of [tex]\(\frac{5}{4}\)[/tex].
Next, we will calculate the slope for each pair of points provided and check which pairs have a slope of [tex]\(\frac{5}{4}\)[/tex].
1. For the points [tex]\((-2, 0)\)[/tex] and [tex]\((2, 5)\)[/tex]:
[tex]\[
\text{slope} = \frac{5 - 0}{2 - (-2)} = \frac{5}{4}
\][/tex]
This pair has a slope of [tex]\(\frac{5}{4}\)[/tex], so it is on a line perpendicular to the original line.
2. For the points [tex]\((-4, 5)\)[/tex] and [tex]\((4, -5)\)[/tex]:
[tex]\[
\text{slope} = \frac{-5 - 5}{4 - (-4)} = \frac{-10}{8} = -\frac{5}{4}
\][/tex]
This pair has a slope of [tex]\(-\frac{5}{4}\)[/tex], not [tex]\(\frac{5}{4}\)[/tex].
3. For the points [tex]\((-3, 4)\)[/tex] and [tex]\((2, 0)\)[/tex]:
[tex]\[
\text{slope} = \frac{0 - 4}{2 - (-3)} = \frac{-4}{5}
\][/tex]
This pair has a slope of [tex]\(-\frac{4}{5}\)[/tex], not [tex]\(\frac{5}{4}\)[/tex].
4. For the points [tex]\((1, -1)\)[/tex] and [tex]\((6, -5)\)[/tex]:
[tex]\[
\text{slope} = \frac{-5 - (-1)}{6 - 1} = \frac{-4}{5}
\][/tex]
This pair has a slope of [tex]\(-\frac{4}{5}\)[/tex], not [tex]\(\frac{5}{4}\)[/tex].
5. For the points [tex]\((2, -1)\)[/tex] and [tex]\((10, 9)\)[/tex]:
[tex]\[
\text{slope} = \frac{9 - (-1)}{10 - 2} = \frac{10}{8} = \frac{5}{4}
\][/tex]
This pair has a slope of [tex]\(\frac{5}{4}\)[/tex].
Summary:
The pairs [tex]\((-2, 0)\)[/tex] and [tex]\((2, 5)\)[/tex], and [tex]\((2, -1)\)[/tex] and [tex]\((10, 9)\)[/tex] have slopes of [tex]\(\frac{5}{4}\)[/tex]. So, the ordered pairs that could be points on a line perpendicular to the original line are:
1. [tex]\((-2, 0)\)[/tex] and [tex]\((2, 5)\)[/tex]
2. [tex]\((2, -1)\)[/tex] and [tex]\((10, 9)\)[/tex]
