Finding Points On Perpendicular Lines A Coordinate Geometry Problem

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In the realm of coordinate geometry, understanding the relationships between lines and points is fundamental. This article delves into a specific problem: given a line MN passing through points (2, 3) and (-3, 2), and a point K at (3, -3), we aim to identify a point that lies on the line perpendicular to MN and passing through K. This exploration will involve calculating slopes, finding equations of lines, and testing point locations. Let's embark on this mathematical journey!

Understanding the Problem: Lines, Points, and Perpendicularity

The coordinate plane serves as our canvas, a two-dimensional space where points are located using ordered pairs (x, y). Lines, in this context, are defined by equations that describe the relationship between x and y coordinates of all points lying on them. A crucial concept here is the slope of a line, which quantifies its steepness and direction. The slope (m) between two points (x1, y1) and (x2, y2) is calculated as:

m = (y2 - y1) / (x2 - x1)

Perpendicular lines hold a special relationship. They intersect at a right angle (90 degrees). A key property of perpendicular lines is that their slopes are negative reciprocals of each other. If a line has a slope m, a line perpendicular to it will have a slope -1/m. This inverse relationship is crucial for solving our problem.

Our specific challenge involves a line MN defined by two points, a separate point K, and a set of potential points. Our mission is to determine which of these potential points lies on the line that is both perpendicular to MN and passes through K. This requires a methodical approach, combining slope calculations, line equation determination, and point verification.

Step-by-Step Solution: Finding the Perpendicular Line

To solve this problem effectively, we'll break it down into a series of logical steps:

1. Calculate the Slope of Line MN

Our first task is to determine the slope of the given line MN, which passes through points (2, 3) and (-3, 2). Using the slope formula:

m_MN = (2 - 3) / (-3 - 2) = (-1) / (-5) = 1/5

Therefore, the slope of line MN is 1/5. This value is the foundation for finding the slope of the perpendicular line.

2. Determine the Slope of the Perpendicular Line

Knowing the slope of MN, we can find the slope of the line perpendicular to it. The perpendicular slope is the negative reciprocal of the original slope:

m_perpendicular = -1 / (1/5) = -5

Thus, the line perpendicular to MN has a slope of -5. This negative reciprocal relationship is the cornerstone of perpendicularity in coordinate geometry.

3. Find the Equation of the Perpendicular Line

Now that we have the slope of the perpendicular line (-5) and a point it passes through (K at (3, -3)), we can determine its equation. We'll use the point-slope form of a linear equation:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope. Plugging in our values:

y - (-3) = -5(x - 3) y + 3 = -5x + 15 y = -5x + 12

This is the equation of the line perpendicular to MN and passing through point K. It defines the relationship between x and y coordinates for all points on this line.

4. Test the Given Points

We are given four potential points: (0, -12), (2, 2), (4, 8), and (5, 13). To determine which point lies on the perpendicular line, we'll substitute the x and y coordinates of each point into the equation y = -5x + 12 and see if the equation holds true.

  • (0, -12): -12 = -5(0) + 12 => -12 = 12 (False)
  • (2, 2): 2 = -5(2) + 12 => 2 = -10 + 12 => 2 = 2 (True)
  • (4, 8): 8 = -5(4) + 12 => 8 = -20 + 12 => 8 = -8 (False)
  • (5, 13): 13 = -5(5) + 12 => 13 = -25 + 12 => 13 = -13 (False)

Only the point (2, 2) satisfies the equation of the perpendicular line. Therefore, this is the point we're looking for.

Detailed Explanation of Key Steps

The Significance of Slope

The slope, as we've seen, is a fundamental concept in understanding lines. It represents the rate of change of the y-coordinate with respect to the x-coordinate. A positive slope indicates an upward trend, while a negative slope signifies a downward trend. The magnitude of the slope reflects the steepness of the line – a larger absolute value means a steeper line.

In our problem, calculating the slope of MN was the crucial first step. It allowed us to determine the orientation of the line and, more importantly, to find the slope of the perpendicular line. The concept of slope is not just a mathematical abstraction; it has real-world applications in areas like engineering, physics, and even economics, where it can represent rates of change in various phenomena.

Point-Slope Form: A Powerful Tool

The point-slope form of a linear equation (y - y1 = m(x - x1)) is a versatile tool for finding the equation of a line when you know its slope and a point it passes through. This form is particularly useful because it directly incorporates the slope and a point, making it easy to construct the equation.

We utilized the point-slope form to find the equation of the perpendicular line. Knowing the slope (-5) and the point K (3, -3), we could directly plug these values into the formula and simplify to obtain the equation y = -5x + 12. This equation is a powerful representation of the line, as it defines all the points that lie on it. Without the point-slope form, finding the equation of the line would have been a more cumbersome process.

Verifying Points: The Final Check

Once we had the equation of the perpendicular line, the final step was to verify which of the given points lay on it. This involved substituting the x and y coordinates of each point into the equation and checking if the equation held true. This process is a fundamental aspect of coordinate geometry, ensuring that a point truly belongs to a particular line or curve.

For the point (2, 2), the substitution resulted in the equation 2 = -5(2) + 12, which simplifies to 2 = 2. This equality confirms that the point (2, 2) lies on the perpendicular line. The other points, when substituted, did not satisfy the equation, indicating that they do not lie on the line. This verification step is crucial for ensuring the accuracy of the solution.

Conclusion: The Point on the Perpendicular

By systematically calculating slopes, determining the equation of the perpendicular line, and verifying the given points, we have successfully identified the point that lies on the line perpendicular to MN and passing through K. Our analysis reveals that the point (2, 2) is the correct answer.

This problem exemplifies the interconnectedness of concepts in coordinate geometry. Understanding slopes, perpendicularity, and linear equations is essential for navigating the coordinate plane and solving geometric problems. The step-by-step approach we employed highlights the importance of breaking down complex problems into manageable parts and applying the appropriate mathematical tools. This exploration not only provides a solution to the specific problem but also reinforces the fundamental principles of coordinate geometry.