- Calculate the slope of the original line through \((3, 4)\) and \((-2, 2)\), which is \(\frac{2}{5} = 0.4\).
- Calculate the slopes of the lines formed by each pair of given points:
- a: \(\frac{2-0}{6-1} = 0.4\)
- b: \(\frac{5-0}{2-0} = 2.5\)
- c: \(\frac{2-0}{8-3} = 0.4\)
- d: \(\frac{-3-(-5)}{-7-(-2)} = -0.4\)
- e: \(\frac{-1-1}{-6-(-1)} = 0.4\)
- Identify that options a, c, and e result in a slope of 0.4, matching the original line.
- The correct options are \(\boxed{\text{a, c, e}}\).
Explanation
- Calculate the slope of the reference line.
The problem asks to identify three pairs of points that lie on lines parallel to the line passing through points \(A(3, 4)\) and \(B(-2, 2)\). Two lines are parallel if they have the same slope.
The formula for the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
First, we calculate the slope of the given line passing through \(A(3, 4)\) and \(B(-2, 2)\):
\[m = \frac{2 - 4}{-2 - 3} = \frac{-2}{-5} = \frac{2}{5} = 0.4\]
Any line parallel to this one must also have a slope of \(0.4\).
- Evaluate the slope for each given option.
We now calculate the slope for each given pair of points and compare it to the reference slope of \(0.4\):
APoints \((1, 0)\) and \((6, 2)\):
\[m_a = \frac{2 - 0}{6 - 1} = \frac{2}{5} = 0.4\]
Since \(m_a = 0.4\), this line is parallel.
BPoints \((0, 0)\) and \((2, 5)\):
\[m_b = \frac{5 - 0}{2 - 0} = \frac{5}{2} = 2.5\]
Since \(m_b
eq 0.4\), this line is not parallel.
CPoints \((3, 0)\) and \((8, 2)\):
\[m_c = \frac{2 - 0}{8 - 3} = \frac{2}{5} = 0.4\]
Since \(m_c = 0.4\), this line is parallel.
DPoints \((-2, -5)\) and \((-7, -3)\):
\[m_d = \frac{-3 - (-5)}{-7 - (-2)} = \frac{2}{-5} = -0.4\]
Since \(m_d
eq 0.4\), this line is not parallel.
EPoints \((-1, 1)\) and \((-6, -1)\):
\[m_e = \frac{-1 - 1}{-6 - (-1)} = \frac{-2}{-5} = 0.4\]
Since \(m_e = 0.4\), this line is parallel.
- Conclusion.
Comparing the results, the lines defined by the point pairs in options a, c, and e all have a slope of \(0.4\), which is identical to the slope of the original line. Therefore, these three pairs lie on lines parallel to the original.
The final answer is \(\boxed{\text{a, c, e}}\).
Examples
Imagine you are designing a city layout and need to build a new road that must be perfectly parallel to an existing main thoroughfare. By finding the coordinates of two markers on the old road, you can determine its slope. Any new construction project along that route, if measured using the same slope, ensures your new road aligns perfectly with the traffic flow, maintaining urban symmetry and efficient planning. For example, if the old road has a slope of \(0.4\), you would orient your new road such that the change in its vertical position divided by the change in its horizontal position is consistently \(\frac{2}{5} = 0.4\).