Finding A Point On A Perpendicular Line A Step By Step Guide

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Determining which point lies on a line that passes through a given point and is perpendicular to another line is a common problem in coordinate geometry. This article will walk you through a step-by-step approach to solve this type of problem, ensuring you understand the underlying concepts and can apply them effectively. We will explore the concepts of slope, perpendicular lines, and the point-slope form of a linear equation.

Understanding the Problem

The question asks us to identify a point that lies on a line. This line has two crucial properties: it passes through a specific point (let's call it H), and it is perpendicular to another line (let's call it FG). To solve this, we'll need to utilize our knowledge of coordinate geometry, specifically the relationships between slopes of perpendicular lines and how to define a line using a point and a slope.

Key Concepts:

  • Slope of a line: The slope (often denoted as m) measures the steepness of a line. It's calculated as the change in the y-coordinate divided by the change in the x-coordinate between two points on the line. The formula for slope is:

    m = (y₂ - y₁) / (x₂ - x₁)

  • Perpendicular lines: Two lines are perpendicular if they intersect at a right angle (90 degrees). A crucial property of perpendicular lines is that their slopes are negative reciprocals of each other. If one line has a slope of m, a line perpendicular to it will have a slope of -1/m.

  • Point-slope form of a linear equation: This form allows us to define a line if we know a point on the line (x₁, y₁) and the slope of the line (m). The point-slope form is:

    y - y₁ = m(x - x₁)

Step-by-Step Solution

Let's break down the solution process into manageable steps:

  1. Determine the slope of line FG: We need the coordinates of points F and G to calculate the slope using the formula m = (y₂ - y₁) / (x₂ - x₁). If these coordinates are not provided in the problem statement, we cannot proceed without that information. Let’s assume for the sake of example that point F is (x₁, y₁) and point G is (x₂, y₂). We would then calculate the slope (m_FG) as:

    m_FG = (y₂ - y₁) / (x₂ - x₁)

    Importance of Accurate Slope Calculation: The slope is the foundation for determining the equation of our perpendicular line. A slight error in calculating the slope of FG will propagate through the rest of the solution, leading to an incorrect answer. Therefore, double-check your calculations and ensure you're using the correct coordinates in the correct order. The slope, often denoted as m, measures the steepness of a line and its direction. It's defined as the change in the y-coordinate (rise) divided by the change in the x-coordinate (run) between any two distinct points on the line. This fundamental concept allows us to quantify the inclination of a line and is crucial for understanding various geometrical relationships. Understanding the slope is crucial because it dictates the direction and steepness of the line. A positive slope indicates an upward trend, while a negative slope indicates a downward trend. The magnitude of the slope reflects the steepness; a larger magnitude means a steeper line. Accurate slope calculation is paramount to solving the problem.

  2. Calculate the slope of the perpendicular line: Once we have the slope of line FG (m_FG), we can find the slope of the line perpendicular to it. The slope of the perpendicular line (m_perp) is the negative reciprocal of m_FG:

    m_perp = -1 / m_FG

    The Negative Reciprocal Relationship: The relationship between the slopes of perpendicular lines is a cornerstone of coordinate geometry. Understanding that perpendicular lines have slopes that are negative reciprocals of each other is key to solving this type of problem. If a line has a slope of m, the slope of any line perpendicular to it is -1/m. This stems from the fact that the product of the slopes of two perpendicular lines is always -1. This negative reciprocal relationship is not just a mathematical rule; it has a geometrical interpretation. When lines are perpendicular, they intersect at a right angle (90 degrees). This geometric property translates into the negative reciprocal relationship between their slopes. The accuracy of finding the negative reciprocal is critical for the rest of the solution.

  3. Use the point-slope form: We know the perpendicular line passes through point H. Let's say the coordinates of point H are (x_H, y_H). We also now know the slope of the perpendicular line (m_perp). We can plug these values into the point-slope form of a linear equation:

y - y_H = m_perp (x - x_H)

**Point-Slope Form for Equation Construction:** The point-slope form is an invaluable tool for constructing the equation of a line when you have a point on the line and its slope. This form directly incorporates the known point (***x₁, y₁***) and the slope (***m***), allowing us to express the linear relationship succinctly. _The point-slope form bridges the connection between a geometric concept (a line defined by a point and a slope) and its algebraic representation (a linear equation)._ The point-slope form serves as a foundation for converting to other forms, such as slope-intercept form (y = mx + b) or standard form (Ax + By = C). This adaptability makes it a versatile tool in coordinate geometry. The point-slope form is particularly useful when dealing with scenarios where a point and a slope are readily available, making it a perfect fit for solving this type of problem. **_Understanding and applying the point-slope form accurately is essential for success._**

  1. Check the given points: Now, we have the equation of the line that passes through point H and is perpendicular to line FG. We can plug in the coordinates of each of the given points into this equation to see if they satisfy it. If a point satisfies the equation, it lies on the line.

    Verifying Points Against the Equation: The final step in solving this problem involves verifying which of the given points lie on the line we've defined. This verification process is a critical check to ensure our solution is correct. We substitute the x and y coordinates of each point into the equation of the line and see if the equation holds true. If both sides of the equation are equal after the substitution, then the point lies on the line. If the equation does not hold true, the point does not lie on the line. This step serves as a quality control measure, confirming that the chosen point aligns with the geometric properties we've established. This is a crucial step to validate your solution and ensure accuracy. The act of plugging in the coordinates and checking for equality is not merely a mechanical step; it's a practical application of the definition of a line in coordinate geometry. A point lies on a line if and only if its coordinates satisfy the equation of that line. Accurate substitution and evaluation are key to successfully identifying the correct point.

Example with Specific Values

Let's assume:

  • Point F is (1, 2)
  • Point G is (3, 6)
  • Point H is (5, 4)
  • The given points to check are:
    • (-6, 10)
    • (-2, -12)
    • (0, -2)
    • (4, 2)

  1. Slope of FG:

    m_FG = (6 - 2) / (3 - 1) = 4 / 2 = 2

  2. Slope of the perpendicular line:

    m_perp = -1 / 2

  3. Equation of the perpendicular line:

    Using point H (5, 4) and m_perp = -1/2: y - 4 = (-1/2)(x - 5) y - 4 = (-1/2)x + 5/2 y = (-1/2)x + 5/2 + 4 y = (-1/2)x + 13/2

  4. Check the given points:

    • (-6, 10): 10 = (-1/2)(-6) + 13/2 10 = 3 + 13/2 10 = 19/2 (False)
    • (-2, -12): -12 = (-1/2)(-2) + 13/2 -12 = 1 + 13/2 -12 = 15/2 (False)
    • (0, -2): -2 = (-1/2)(0) + 13/2 -2 = 13/2 (False)
    • (4, 2): 2 = (-1/2)(4) + 13/2 2 = -2 + 13/2 2 = 9/2 (False)

In this specific example, none of the provided points lie on the perpendicular line. This illustrates the importance of carefully performing each step and verifying the final result.

Common Pitfalls and How to Avoid Them

  • Incorrectly calculating the slope: Always double-check the coordinates and ensure you're using the correct formula (change in y divided by change in x). A mistake here will cascade through the rest of the solution.
  • Forgetting the negative reciprocal: When finding the slope of a perpendicular line, remember to take the negative reciprocal, not just the reciprocal. Dropping the negative sign is a common error.
  • Algebraic mistakes: Carefully perform algebraic manipulations when simplifying equations. A small arithmetic error can lead to an incorrect equation and, consequently, the wrong answer.
  • Misunderstanding point-slope form: Ensure you're plugging the correct values into the point-slope formula (the point the line passes through and the slope of the line).

By understanding the concepts, following the steps carefully, and avoiding common pitfalls, you can confidently solve problems involving perpendicular lines in coordinate geometry.

Conclusion

Finding a point on a line perpendicular to another involves understanding the relationship between slopes of perpendicular lines and the point-slope form of a line equation. By calculating the slope of the given line, finding its negative reciprocal, constructing the equation of the perpendicular line, and then checking the given points against the equation, we can accurately determine the solution. Careful attention to detail and a solid grasp of the underlying concepts are crucial for success in these types of problems.