Problem 81
Question
Determine whether the lines \(L_{1}\) and \(L_{2}\) passing through the pairs of points are parallel, perpendicular, or neither.$$\begin{aligned}&L_{1}:(-2,-2),(2,10)\\\&L_{2}:(-1,3),(3,9)\end{aligned}$$
Step-by-Step Solution
Verified Answer
The lines \(L_{1}\) and \(L_{2}\) are parallel.
1Step 1: Calculate the slope of \(L_{1}\)
The slope of a line passing through points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) is given by \(\frac{y_{2} - y_{1}}{x_{2} - x_{1}}\). Therefore, the slope of \(L_{1}\) passing through points (-2, -2) and (2, 10) can be calculated as \(\frac{10 - (-2)}{2 - (-2)} = 3\)
2Step 2: Calculate the slope of \(L_{2}\)
Similarly, the slope of \(L_{2}\) passing through points (-1, 3) and (3, 9) can be calculated as \(\frac{9 - 3}{3 - (-1)} = 3\)
3Step 3: Compare the slopes and deduce the relationship
Since the slopes of both lines are equal, \(L_{1}\) and \(L_{2}\) are parallel
Key Concepts
Slope of a LineParallel LinesPerpendicular Lines
Slope of a Line
Understanding the slope of a line is crucial for analyzing the relationships between lines on a graph. The slope is a measure of how steep a line is, defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. Mathematically, it's expressed as \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
When calculating the slope, it's important to consistently use the same order for subtracting coordinates to avoid errors. For instance, if you start with the second y-coordinate subtracted from the first one, do the same with the x-coordinates. The slope can be positive, negative, zero, or undefined. A positive slope indicates an upward trend from left to right, while a negative slope indicates a downward trend. A zero slope means the line is horizontal, while an undefined slope (when the line is vertical) occurs when the denominator in the slope formula is zero.
When calculating the slope, it's important to consistently use the same order for subtracting coordinates to avoid errors. For instance, if you start with the second y-coordinate subtracted from the first one, do the same with the x-coordinates. The slope can be positive, negative, zero, or undefined. A positive slope indicates an upward trend from left to right, while a negative slope indicates a downward trend. A zero slope means the line is horizontal, while an undefined slope (when the line is vertical) occurs when the denominator in the slope formula is zero.
Parallel Lines
Parallel lines are lines in the same plane that never intersect, no matter how far they are extended. The fundamental property of parallel lines is that they have the same slope. Thus, in analytical geometry, determining if two lines are parallel involves comparing their slopes.
For example, if two lines have slopes that are equal, as verified using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), then these lines are considered parallel. It's essential, especially when solving problems involving equations of lines, to ensure that the slopes are computed accurately to conclude their parallelism. Incorrect calculation can lead to a misinterpretation of the lines' relationship.
For example, if two lines have slopes that are equal, as verified using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), then these lines are considered parallel. It's essential, especially when solving problems involving equations of lines, to ensure that the slopes are computed accurately to conclude their parallelism. Incorrect calculation can lead to a misinterpretation of the lines' relationship.
Perpendicular Lines
Perpendicular lines intersect at a right angle and are associated with the concept of orthogonality. The slopes of two perpendicular lines are not just any pair of numbers but are specifically negative reciprocals of each other. If one line has a slope of \(m\), the slope of a line perpendicular to it must be \(\frac{-1}{m}\), provided \(m\) is not zero.
For instance, if one line has a slope of 3, a line perpendicular to it will have a slope of \(\frac{-1}{3}\). This property is key for solving geometry problems and analyzing figures on a coordinate plane. To verify perpendicularity, calculate the slopes of the two lines in question and check if the product of the slopes is \( -1 \). Remember, vertical and horizontal lines are also perpendicular, with their slopes being undefined and zero, respectively.
For instance, if one line has a slope of 3, a line perpendicular to it will have a slope of \(\frac{-1}{3}\). This property is key for solving geometry problems and analyzing figures on a coordinate plane. To verify perpendicularity, calculate the slopes of the two lines in question and check if the product of the slopes is \( -1 \). Remember, vertical and horizontal lines are also perpendicular, with their slopes being undefined and zero, respectively.
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