Finding A Point On A Parallel Line A Step-by-Step Guide
In mathematics, determining points on a parallel line is a fundamental concept in coordinate geometry. This article provides a comprehensive guide on how to find a point on a line parallel to a given line that passes through a specified point. Understanding this concept is essential for various mathematical applications and problem-solving scenarios. Let's explore the step-by-step process and underlying principles to master this skill.
Understanding Parallel Lines
Parallel lines are lines in a plane that never intersect. This crucial characteristic dictates their slopes; parallel lines have the same slope. To find a point on a line parallel to a given line, we must first understand how to determine the slope of the given line. The slope, often denoted as m, represents the steepness and direction of a line. It is calculated using the formula:
m = (y₂ - y₁) / (x₂ - x₁)
where (x₁, y₁) and (x₂, y₂) are two points on the line. Once we know the slope of the original line, we know the slope of any line parallel to it. The next step is to use this slope along with the given point through which the parallel line passes to find other points on the new line.
Determining the Slope of Line KL
To determine which point could be on the line parallel to line KL, we first need to calculate the slope of line KL. Let's assume we have two points K(x₁, y₁) and L(x₂, y₂) on line KL. The slope (m) of line KL is given by:
m = (y₂ - y₁) / (x₂ - x₁)
Suppose K is at (-2, 3) and L is at (4, -1). Plugging these values into the formula, we get:
m = (-1 - 3) / (4 - (-2)) = -4 / 6 = -2/3
Thus, the slope of line KL is -2/3. Any line parallel to KL will also have a slope of -2/3. This understanding is critical in our subsequent steps to identify a point on the parallel line.
Finding the Equation of the Parallel Line
Now that we know the slope of the parallel line, which is -2/3, and we have a point M through which the line passes, we can use the point-slope form of a linear equation to find the equation of the parallel line. The point-slope form is given by:
y - y₁ = m(x - x₁)
where (x₁, y₁) is the point M and m is the slope. Let's assume point M is at (2, 1). Substituting these values into the point-slope form, we get:
y - 1 = -2/3 (x - 2)
To simplify, we can convert this equation to slope-intercept form (y = mx + b) where b is the y-intercept:
y - 1 = -2/3x + 4/3 y = -2/3x + 4/3 + 1 y = -2/3x + 7/3
This equation represents the line parallel to KL and passing through point M. Now, we can use this equation to test which of the given points lies on this line.
Testing the Given Points
We are given the following points: (-10, 0), (-6, 2), (0, -6), and (8, -10). To determine which point lies on the parallel line, we substitute the x and y coordinates of each point into the equation y = -2/3x + 7/3 and check if the equation holds true.
Testing Point (-10, 0)
Substitute x = -10 and y = 0 into the equation:
0 = -2/3(-10) + 7/3 0 = 20/3 + 7/3 0 = 27/3 0 = 9
This statement is false, so the point (-10, 0) does not lie on the line.
Testing Point (-6, 2)
Substitute x = -6 and y = 2 into the equation:
2 = -2/3(-6) + 7/3 2 = 12/3 + 7/3 2 = 4 + 7/3 2 = 12/3 + 7/3 2 = 19/3
This statement is also false, so the point (-6, 2) does not lie on the line.
Testing Point (0, -6)
Substitute x = 0 and y = -6 into the equation:
-6 = -2/3(0) + 7/3 -6 = 0 + 7/3 -6 = 7/3
This statement is false, indicating that the point (0, -6) does not lie on the line.
Testing Point (8, -10)
Substitute x = 8 and y = -10 into the equation:
-10 = -2/3(8) + 7/3 -10 = -16/3 + 7/3 -10 = -9/3 -10 = -3
This statement is false as well, meaning the point (8, -10) is not on the line. After reviewing our calculations, there appears to be an error, let's backtrack to the equation of the parallel line: y = -2/3x + 7/3. It’s crucial to ensure accuracy here.
Let's re-test the points with the equation of the parallel line, making sure each step is carefully checked.
Re-testing Point (-10, 0)
Substitute x = -10 and y = 0 into the equation y = -2/3x + 7/3:
0 = -2/3(-10) + 7/3 0 = 20/3 + 7/3 0 = 27/3 0 = 9
This remains false. Point (-10, 0) is still not on the line.
Re-testing Point (-6, 2)
Substitute x = -6 and y = 2 into the equation:
2 = -2/3(-6) + 7/3 2 = 4 + 7/3 2 = 12/3 + 7/3 2 = 19/3
Again, this is false. Point (-6, 2) is not on the line.
Re-testing Point (0, -6)
Substitute x = 0 and y = -6 into the equation:
-6 = -2/3(0) + 7/3 -6 = 0 + 7/3 -6 = 7/3
This is still false, confirming that the point (0, -6) is not on the line.
Re-testing Point (8, -10)
Substitute x = 8 and y = -10 into the equation:
-10 = -2/3(8) + 7/3 -10 = -16/3 + 7/3 -10 = -9/3 -10 = -3
This also remains false. Thus, none of the provided points appear to lie on the calculated parallel line. It's possible there was an initial error in the example points selected for K, L, or M, or in the provided answer choices. To proceed definitively, we need to either re-evaluate the initial points or consider different points.
An Alternative Approach: Using the Slope and a Point
If we reconsider the original question more generically, we can affirm the basic methodology. We calculated the slope m as -2/3 and assumed point M as (2, 1). If we were to find another point on the line parallel to KL passing through M, we can use the slope to "move" from point M to another point on the line. This involves understanding the rise over run associated with the slope.
Given the slope is -2/3, for every 3 units we move horizontally (run), we move -2 units vertically (rise). Starting from point M (2, 1), we can apply this:
- Move 3 units to the right (x + 3): 2 + 3 = 5
- Move 2 units down (y - 2): 1 - 2 = -1
Thus, a new point on the line would be (5, -1). We can continue this process to find multiple points on the line. This approach reaffirms the importance of understanding the slope in determining points on a line.
Conclusion
Finding a point on a line parallel to a given line involves several key steps: determining the slope of the original line, using the same slope for the parallel line, applying the point-slope form to find the equation of the parallel line, and then testing points to see if they satisfy the equation. While the initial example points did not result in a match with the provided options, the methodology remains robust. By understanding and applying these principles, one can effectively solve such problems in coordinate geometry. Accuracy in calculations and a clear understanding of the underlying concepts are vital for success in these types of mathematical challenges. Remember, practice is crucial to mastering these skills, and revisiting the basics can often help clarify any confusion that arises during problem-solving.
