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Sagot :
To determine which ordered pairs could be points on a line that is perpendicular to a given line with a slope of [tex]\(-\frac{4}{5}\)[/tex], we need to understand the relationship between the slopes of perpendicular lines. Specifically, the slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line.
1. Identify the slope of the perpendicular line:
The slope of the given line is [tex]\(-\frac{4}{5}\)[/tex]. The negative reciprocal of [tex]\(-\frac{4}{5}\)[/tex] is obtained by flipping the fraction and changing the sign:
[tex]\[ \text{Perpendicular slope} = -\left(-\frac{5}{4}\right) = \frac{5}{4} \][/tex]
2. Check each pair of points to see if they yield this slope:
- 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 the correct slope of [tex]\(\frac{5}{4}\)[/tex].
- 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 does not have the slope we're looking for.
- 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 does not have the slope we're looking for.
- 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 does not have the slope we're looking for.
- 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 the correct slope of [tex]\(\frac{5}{4}\)[/tex].
So, the ordered pairs that could be points on a line perpendicular to the line with a slope of [tex]\(-\frac{4}{5}\)[/tex] are:
- [tex]\((-2, 0)\)[/tex] and [tex]\((2, 5)\)[/tex]
- [tex]\((2, -1)\)[/tex] and [tex]\((10, 9)\)[/tex]
1. Identify the slope of the perpendicular line:
The slope of the given line is [tex]\(-\frac{4}{5}\)[/tex]. The negative reciprocal of [tex]\(-\frac{4}{5}\)[/tex] is obtained by flipping the fraction and changing the sign:
[tex]\[ \text{Perpendicular slope} = -\left(-\frac{5}{4}\right) = \frac{5}{4} \][/tex]
2. Check each pair of points to see if they yield this slope:
- 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 the correct slope of [tex]\(\frac{5}{4}\)[/tex].
- 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 does not have the slope we're looking for.
- 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 does not have the slope we're looking for.
- 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 does not have the slope we're looking for.
- 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 the correct slope of [tex]\(\frac{5}{4}\)[/tex].
So, the ordered pairs that could be points on a line perpendicular to the line with a slope of [tex]\(-\frac{4}{5}\)[/tex] are:
- [tex]\((-2, 0)\)[/tex] and [tex]\((2, 5)\)[/tex]
- [tex]\((2, -1)\)[/tex] and [tex]\((10, 9)\)[/tex]
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