To determine which ordered pairs could be points on a line that is perpendicular to a line with a given slope, we need to find the slope of the perpendicular line. The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope.
Given slope of the original line:
[tex]\[ m = -\frac{4}{5} \][/tex]
The negative reciprocal of this slope is:
[tex]\[ m_{\text{perpendicular}} = -\frac{1}{-\frac{4}{5}} = \frac{5}{4} \][/tex]
Next, we need to check each pair of points to see if they form a line with a slope of [tex]\(\frac{5}{4}\)[/tex].
1. Points [tex]\((-2, 0)\)[/tex] and [tex]\((2, 5)\)[/tex]:
[tex]\[
\text{slope} = \frac{5 - 0}{2 - (-2)} = \frac{5}{4}
\][/tex]
This pair forms a line with the correct slope of [tex]\(\frac{5}{4}\)[/tex].
2. 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 does not form a line with the correct slope.
3. Points [tex]\((-3, 4)\)[/tex] and [tex]\((2, 0)\)[/tex]:
[tex]\[
\text{slope} = \frac{0 - 4}{2 - (-3)} = \frac{-4}{5}
\][/tex]
This pair does not form a line with the correct slope.
4. Points [tex]\((1, -1)\)[/tex] and [tex]\((6, -5)\)[/tex]:
[tex]\[
\text{slope} = \frac{-5 - (-1)}{6 - 1} = \frac{-4}{5}
\][/tex]
This pair does not form a line with the correct slope.
5. 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 forms a line with the correct slope of [tex]\(\frac{5}{4}\)[/tex].
Hence, the ordered pairs that could be points on a line perpendicular to the one with slope [tex]\(-\frac{4}{5}\)[/tex] are:
1. [tex]\((-2, 0)\)[/tex] and [tex]\((2, 5)\)[/tex]
2. [tex]\((2, -1)\)[/tex] and [tex]\((10, 9)\)[/tex]
