Verified Answer :
- Identify potential lines by checking the collinearity of the provided points.
- Two sets of collinear points are identified: \((-4, 2), (-2, 2), (4, 2)\) lying on \(y = 2\) and \((-2, 2), (-1, 3), (-5, -1)\) lying on \(y = x + 4\).
- Select the horizontal line \(y = 2\) as the intended path, confirming that points \((-4, 2), (-2, 2),\) and \((4, 2)\) satisfy the equation.
- The final points are \(\boxed{(-4, 2), (-2, 2), (4, 2)}\).
Explanation
- Problem Analysis and Coordinate Geometry Basics
To determine which points lie on a line passing through a point \(P\) and parallel to a given line, we first analyze the properties of lines in a 2D coordinate system. A line in the coordinate plane can be described by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Two lines are parallel if and only if they have the same slope \(m\). If we have a point \(P(x_p, y_p)\) and a slope \(m\), the equation of the line is given by the point-slope form: \(y - y_p = m(x - x_p)\). Given the set of points \((-4, 2), (-1, 3), (-2, 2), (4, 2), (-5, -1)\), we identify potential sets of collinear points that could represent such a line.
- Identifying Collinear Sets and Lines
We observe two groups of points that are collinear.
First, consider the points \(A(-4, 2)\), \(C(-2, 2)\), and \(D(4, 2)\). These points all share the same y-coordinate, \(y = 2\). This represents a horizontal line with a slope \(m = 0\). If the point \(P\) is one of these points and the given line is any horizontal line (such as the x-axis, \(y=0\)), the required line is \(y=2\). Any point on this line must satisfy \(y=2\). Thus, the points \((-4, 2), (-2, 2),\) and \((4, 2)\) lie on this line.
Second, consider the points \(C(-2, 2)\), \(B(-1, 3)\), and \(E(-5, -1)\). The slope between \(C\) and \(B\) is \(\frac{3-2}{-1-(-2)} = \frac{1}{1} = 1\). The slope between \(B\) and \(E\) is \(\frac{-1-3}{-5-(-1)} = \frac{-4}{-4} = 1\). Since the slopes are equal, these points lie on a line with slope \(m=1\). Using the point-slope form with point \((-1, 3)\), we get \(y - 3 = 1(x - (-1))\), which simplifies to \(y = x + 4\). These points \((-2, 2), (-1, 3),\) and \((-5, -1)\) also form a valid line.
We choose the line \(y = 2\) for our primary example.
- Verification and Final Selection
Assuming the line is defined by \(y = 2\), we test each of the provided options:
- \((-4, 2)\): \(2 = 2\), which is true.
- \((-1, 3)\): \(3 = 2\), which is false.
- \((-2, 2)\): \(2 = 2\), which is true.
- \((4, 2)\): \(2 = 2\), which is true.
- \((-5, -1)\): \(-1 = 2\), which is false.
The three points that lie on this line are \((-4, 2), (-2, 2),\) and \((4, 2)\). Therefore, these are the selected options.
Examples
In urban planning, identifying collinear points is essential for infrastructure development. For example, if you need to place a series of streetlights along a straight road segment, you are effectively finding points that lie on a specific line. By determining the equation of the road (the line) based on known landmarks (points), city planners ensure that utilities are aligned correctly, maximizing efficiency and minimizing the cost of installation. This geometric approach is a fundamental tool for engineers and planners to organize space effectively.