Verified Answer :
- Calculate the required perpendicular slope: Given \(m_1 = -4/5\), the perpendicular slope is \(m_2 = 5/4\).
- Apply the slope formula \(m = (y_2 - y_1) / (x_2 - x_1)\) to each point pair.
- Identify the pairs \((-2, 0)\) and \((2, 5)\), and \((2, -1)\) and \((10, 9)\) as having the correct slope of \(\frac{5}{4}\).
- Conclusion: The two pairs are \((-2, 0)\) and \((2, 5)\); \((2, -1)\) and \((10, 9)\).
Explanation
- Analyze the condition for perpendicular lines.
The goal is to identify which two pairs of points define a line perpendicular to a line with a slope of \(m_1 = -\frac{4}{5}\).
Two lines are perpendicular if the product of their slopes is \(-1\). Let \(m_2\) be the slope of the perpendicular line. Then we must have:
\[m_1 \cdot m_2 = -1\]
\[-\frac{4}{5} \cdot m_2 = -1\]
\[m_2 = -1 \cdot \left( -\frac{5}{4} \right) = \frac{5}{4}\]
So, we are looking for lines with a slope of \(\frac{5}{4}\). The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
We will test each pair of points to see which results in a slope of \(\frac{5}{4}\).
- Calculate the slope for each option and compare.
For each pair of points, we compute the slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
- \((-2, 0)\) and \((2, 5)\):
\[m = \frac{5 - 0}{2 - (-2)} = \frac{5}{4}\]
This slope is \(\frac{5}{4}\), so this pair is correct.
- \((-4, 5)\) and \((4, -5)\):
\[m = \frac{-5 - 5}{4 - (-4)} = \frac{-10}{8} = -\frac{5}{4}\]
This slope is \(-\frac{5}{4}\), which is not \(\frac{5}{4}\).
- \((-3, 4)\) and \((2, 0)\):
\[m = \frac{0 - 4}{2 - (-3)} = \frac{-4}{5}\]
This slope is \(-\frac{4}{5}\), which is not \(\frac{5}{4}\).
- \((1, -1)\) and \((6, -5)\):
\[m = \frac{-5 - (-1)}{6 - 1} = \frac{-4}{5}\]
This slope is \(-\frac{4}{5}\), which is not \(\frac{5}{4}\).
- \((2, -1)\) and \((10, 9)\):
\[m = \frac{9 - (-1)}{10 - 2} = \frac{10}{8} = \frac{5}{4}\]
This slope is \(\frac{5}{4}\), so this pair is correct.
- Conclusion of findings.
Based on our calculations, the lines formed by the pairs \((-2, 0)\) and \((2, 5)\) and the pairs \((2, -1)\) and \((10, 9)\) both have a slope of \(\frac{5}{4}\). Therefore, these are the two options that represent lines perpendicular to the line with a slope of \(-\frac{4}{5}\).
The two correct options are:
- \((-2, 0)\) and \((2, 5)\)
- \((2, -1)\) and \((10, 9)\)
Examples
In civil engineering, determining if two structural beams are perpendicular is crucial for stability. If one beam is represented by a line on a blueprint with a specific slope, engineers can use the negative reciprocal rule to calculate the required slope for a supporting crossbeam. Ensuring these slopes are correctly calculated guarantees the intersection is exactly at a 90-degree angle, maintaining the structural integrity of the entire construction.