- Identify the perpendicular slope requirement: \(- \frac{4}{5} \cdot m_2 = -1 \implies m_2 = \frac{5}{4}\).
- Calculate the slope for each given pair using \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
- Identify the pairs \((-2, 0)\) and \((2, 5)\) and \((2, -1)\) and \((10, 9)\) as having the required slope of \(\frac{5}{4}\).
- Final Answer: \(\boxed{\text{(-2, 0) and (2, 5), (2, -1) and (10, 9)}}\)
Explanation
- Understanding the Perpendicular Slope Condition
To determine which pairs of points lie on a line perpendicular to a given line with a slope of \(-\frac{4}{5}\), we first recall the condition for perpendicular lines. Two lines are perpendicular if and only if the product of their slopes is \(-1\). Let \(m_1\) be the slope of the given line and \(m_2\) be the slope of the perpendicular line. Thus, \(m_1 \cdot m_2 = -1\). Given \(m_1 = -\frac{4}{5}\), we have \(-\frac{4}{5} \cdot m_2 = -1\), which implies \(m_2 = \frac{5}{4}\). We are looking for two pairs of points that result in a slope of \(\frac{5}{4}\). The slope formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
- Evaluating the Slopes of the Given Pairs
We now test the slope for each given pair of points:
- For \((-2, 0)\) and \((2, 5)\):
\[m = \frac{5 - 0}{2 - (-2)} = \frac{5}{2 + 2} = \frac{5}{4}\]
This pair has a slope of \(\frac{5}{4}\).
- For \((-4, 5)\) and \((4, -5)\):
\[m = \frac{-5 - 5}{4 - (-4)} = \frac{-10}{8} = -\frac{5}{4}\]
This slope is \(-\frac{5}{4}\), not \(\frac{5}{4}\).
- For \((-3, 4)\) and \((2, 0)\):
\[m = \frac{0 - 4}{2 - (-3)} = \frac{-4}{5}\]
This slope is \(-\frac{4}{5}\), not \(\frac{5}{4}\).
- For \((1, -1)\) and \((6, -5)\):
\[m = \frac{-5 - (-1)}{6 - 1} = \frac{-4}{5}\]
This slope is \(-\frac{4}{5}\), not \(\frac{5}{4}\).
- For \((2, -1)\) and \((10, 9)\):
\[m = \frac{9 - (-1)}{10 - 2} = \frac{10}{8} = \frac{5}{4}\]
This pair has a slope of \(\frac{5}{4}\).
- Conclusion
Comparing the slopes calculated in the previous step, we see that the pairs \((-2, 0)\) and \((2, 5)\) and \((2, -1)\) and \((10, 9)\) both have a slope of \(\frac{5}{4}\). These are the lines that are perpendicular to the line with a slope of \(-\frac{4}{5}\). Thus, the two correct options are the first and the last pairs.
Examples
In construction and engineering, ensuring that walls or supporting beams are perfectly perpendicular is crucial for structural integrity. This is often achieved by ensuring slopes or gradients of intersecting components are negative reciprocals of each other, as shown here. For example, if a roof slope is determined by a specific grade, support rafters must be calculated using this perpendicular slope principle to ensure they fit correctly and provide stable support.