Problem 82 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the lines \(L_{1}\) and \(L_{2}\) passing through the pairs of points are parallel, perpendicular, or neither.$$\begin{aligned}&L_{1}:(-1,-7),(4,3)\\\&L_{2}:(1,5),(-2,-7)\end{aligned}$$.

Step-by-Step Solution

Answer

The lines \(L_{1}\) and \(L_{2}\) passing through the pairs of points are neither parallel nor perpendicular.

Step 1: Calculate the Slope of Line \(L_{1}\)

First, calculate the slope of line \(L_{1}\) using the formula for the slope of a line, \(m = (y_{2} - y_{1})/(x_{2} - x_{1})\), substituting the given points \(-1, -7\), and \(4, 3\). Hence, \(m_{1} = (3 - (-7))/(4 - (-1)) = (3 + 7) / (4 + 1) = 10/5 = 2\). Therefore, the slope of line \(L_{1}\) is 2.

Step 2: Calculate the Slope of Line \(L_{2}\)

Next, calculate the slope of line \(L_{2}\) using the same formula, substituting the given points \(1, 5\) and \(-2, -7\). therefore, \(m_{2} = (-7 - 5)/(-2 - 1) = -12/-3 = 4\). Thus, the slope of line \(L_{2}\) is 4.

Step 3: Determine Relationship between Lines

Analyze the relationship between the slopes. They are not equal, so the lines are not parallel. Also they are not negative reciprocals of each other, so the lines are not perpendicular. Therefore, the lines \(L_{1}\) and \(L_{2}\) are neither parallel nor perpendicular.

Key Concepts

Slope of a Line

Understanding the slope of a line is critical in analyzing its direction and steepness. The slope expresses the ratio of the vertical change to the horizontal change between two points on the line. In mathematical terms, the slope, often represented as m, is calculated using the formula m = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.

Keep in mind that if the slope is positive, the line rises as it moves from left to right. Conversely, if the slope is negative, the line falls as it moves from left to right. A zero slope means the line is horizontal, while an undefined slope (division by zero) indicates a vertical line. Consequently, an important step in examining the relationship between two lines involves comparing their slopes.

Parallel Lines

Parallel lines are lines in the same plane that never intersect, meaning they have the same slope. This property is essential when analyzing geometric figures and solving various algebraic problems. For example, in a coordinate plane, if two lines L₁ and L₂ have equations y = m₁x + b₁ and y = m₂x + b₂ respectively, and m₁ = m₂, then those lines are considered parallel.

Hence, always compare the slopes of two lines to determine if they are parallel. If their slopes are equal and they have different y-intercepts, they will not meet, illustrating the concept of parallel lines in coordinate geometry.

Perpendicular Lines

When two lines in a plane intersect at a right angle, they are said to be perpendicular. The slopes of perpendicular lines have a particular relationship: they are negative reciprocals of each other. To put it simply, if one line has a slope of m, the line perpendicular to it will have a slope of -1/m.

You can utilize this rule to solve geometric problems or confirm perpendicularity in a coordinate plane. For any two lines with slopes m₁ and m₂, if the product m₁ * m₂ = -1, the lines are confirmed to be perpendicular. This property is especially helpful in constructing or verifying right angles within geometrical shapes.

Coordinate Geometry

Coordinate geometry, or analytic geometry, allows us to analyze geometrical shapes using algebra and the Cartesian coordinate system. This branch of mathematics is crucial in determining the relationships and properties of figures like lines, circles, and polygons by representing them with equations and plotting them on the coordinate plane.

In the context of lines, coordinate geometry enables us to express their positions using linear equations and calculate important attributes such as length, slope, and intersections. This system is fundamental when dealing with geometric problems, enabling the translation of visual information into algebraic expressions and vice versa, bridging the gap between algebra and geometry.