- Identify the required perpendicular slope as the negative reciprocal of the given slope: \(m_2 = -\frac{1}{-4/5} = 5/4\).
- Calculate the slope for each provided pair of points using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
- Compare the calculated slopes with the required perpendicular slope of \(1.25\).
- Identify that the pair \((2, -1)\) and \((10, 9)\) yields a slope of \(1.25\), identifying it as the correct segment. \(\boxed{(2, -1) \text{ and } (10, 9)}\)
Explanation
- Determine the target slope and formula.
To find a line perpendicular to a given line with a slope of \(m_1 = -\frac{4}{5}\), we must determine the required perpendicular slope \(m_2\). Two lines are perpendicular if the product of their slopes is \(-1\). Thus, we solve for \(m_2\):
\[m_1 \cdot m_2 = -1\]
\[-\frac{4}{5} \cdot m_2 = -1\]
\[m_2 = \frac{-1}{-\frac{4}{5}} = \frac{5}{4} = 1.25\]
Any line segment passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) with a slope of \(1.25\) will be perpendicular to the original line. The slope formula is:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
We evaluate the slope for each given pair of points to see which ones equal \(1.25\).
- Evaluate the slope for each pair of points.
We test each pair:
- For \((-4, 5)\) and \((4, -5)\): \(m = \frac{-5 - 5}{4 - (-4)} = \frac{-10}{8} = -1.25\) (Incorrect)
- For \((1, -1)\) and \((6, -5)\): \(m = \frac{-5 - (-1)}{6 - 1} = \frac{-4}{5} = -0.8\) (Incorrect)
- For \((2, -1)\) and \((10, 9)\): \(m = \frac{9 - (-1)}{10 - 2} = \frac{10}{8} = 1.25\) (Correct)
- For \((-2, 0)\) and \((3, 4)\): \(m = \frac{4 - 0}{3 - (-2)} = \frac{4}{5} = 0.8\) (Incorrect)
- For \((-3, 4)\) and \((2, 0)\): \(m = \frac{0 - 4}{2 - (-3)} = \frac{-4}{5} = -0.8\) (Incorrect)
Upon re-evaluating the options provided, only one pair \((2, -1)\) and \((10, 9)\) yields the perpendicular slope of \(1.25\). It appears there is a constraint in the question format, but based on the calculations, only this specific pair qualifies as a line perpendicular to the original line.
- Conclusion.
The only pair of points that form a line perpendicular to the original line (slope \(-4/5\)) is the pair \((2, -1)\) and \((10, 9)\), as it results in the perpendicular slope \(m = 1.25\).
\(\boxed{(2, -1) \text{ and } (10, 9)}\)
Examples
In architectural drafting, perpendicular lines are essential for creating corners and structural supports. If a structural beam follows a path with a slope of \(-\frac{4}{5}\), a cross-beam must follow a path with a slope of \(\frac{5}{4}\) to be perfectly perpendicular, ensuring the structural integrity of the frame. This calculation helps ensure that all angles are precisely 90 degrees in complex designs.