Finding the Value of K for Parallel Lines | Geometry and Slope Equations
Parallel lines are a fundamental concept in geometry, where lines that never meet and have the same slope are of great interest. In this article, we will explore the process of finding the value of K when two lines are parallel. We will delve into the mathematical calculations involved and provide a step-by-step approach to solving similar problems.
Understanding Slope and Parallel Lines
In geometry, the slope of a line is a measure of its steepness, defined as the rise over the run. For two lines to be parallel, their slopes must be equal. If the equation of one line is given in the form 5y - 2x - 7 0, we can rearrange this to find its slope.
Deriving the Slope of the Given Line
The equation of the given line is 5y - 2x - 7 0. To find its slope, we can rearrange it into the slope-intercept form, y mx b, where m is the slope.
$5y - 2x - 7 0$
$5y 2x 7$
$y frac{2}{5}x frac{7}{5}$
The slope m is frac{2}{5}.
Finding the Value of K
Given points A(3, K) and B(2, -5), we need to find the value of K such that the line passing through these points is parallel to the given line. For two lines to be parallel, the slopes must be equal. Let's calculate the slope of the line passing through points A and B.
$m frac{K - (-5)}{3 - 2}$
$m K 5$
Since the lines are parallel, their slopes are equal:
$K 5 -frac{2}{5}$
Rearranging the equation:
$K 5 -frac{2}{5}$
$K -frac{2}{5} - 5$
$K -frac{2}{5} - frac{25}{5}$
$K -frac{27}{5}$
$K -5.4$
Step-by-Step Verification
We can verify our solution by substituting K -5.4 back into the equation of the line through points A and B. The line passing through B(2, -5) and A(3, K) should be parallel to the given line.
Equation of the line through B and A:
$y - (-5) -frac{2}{5}(x - 2)$
$y 5 -frac{2}{5}x frac{4}{5}$
$5y 25 -2x 4$
$5y -2x - 21$
$y -frac{2}{5}x - frac{21}{5}$
The slope of this line is frac{-2}{5}, which is the same as the given line.
Conclusion
Understanding the concept of slope and parallel lines is crucial for solving geometric problems. By following the step-by-step approach outlined in this article, you can find the value of K for parallel lines. This process enhances your problem-solving skills and provides a clear understanding of the relationship between the slopes of parallel lines.