Problem 48 - Elementary Algebra Within Reach

Question

In Exercises 46-49, determine whether the lines \(L_{1}\) and \(L_{2}\) that pass through the pairs of points are parallel, perpendicular, or neither. $$ \begin{aligned} &L_{1}:(12,0),(7,-2) \\ &L_{2}:(0,7),(-5,9) \end{aligned} $$

Step-by-Step Solution

Answer

The lines \(L_{1}\) and \(L_{2}\) are neither parallel nor perpendicular.

Step 1: Compute slope of \(L_{1}\)

The slope \(m_{1}\) of a line passing through two points \((x_{1},y_{1})\) and \((x_{2},y_{2})\) is given by \(m_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\). For \(L_{1}\), the points are \((12,0)\) and \((7,-2)\), so the slope \(m_{1}\) is given by \(m_{1}=\frac{-(2)-(0)}{7-12}=-\frac{2}{-5}=0.4\).

Step 2: Compute slope of \(L_{2}\)

Using the same formula, we can find the slope \(m_{2}\) of \(L_{2}\), which passes through points \((0,7)\) and \((-5,9)\). So, \(m_{2}=\frac{9-7}{-5-0}=\frac{2}{-5}=-0.4.\)

Step 3: Compare the slopes

Now comparing the slopes \(m_{1}\) and \(m_{2}\) determined in steps 1 and 2: \(m_{1}=0.4\) and \(m_{2}=-0.4.\) Since these slopes are not equal nor are they negative reciprocals of each other, the lines \(L_{1}\) and \(L_{2}\) are neither parallel nor perpendicular.

Key Concepts

Slope Calculation

Calculating the slope of a line is a fundamental concept in geometry. The slope, often denoted as \( m \), represents how steep a line is and the direction it goes. To find the slope of a line through two points, let's say \((x_1, y_1)\) and \((x_2, y_2)\), we use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula calculates the change in \( y \) values divided by the change in \( x \) values. It essentially measures how much \( y \) changes for each unit change in \( x \). If the result is positive, the line slopes upwards from left to right. If negative, it slopes downwards. This method of calculation helps us understand the shape and direction of a line in a coordinate system.

Negative Reciprocal

The concept of negative reciprocals is essential when determining if two lines are perpendicular. In simple terms, if one line has a slope of \( m \), a line perpendicular to it will have a slope of \( -\frac{1}{m} \). This means you take the reciprocal of the slope and then change its sign.

• A slope of \( 2 \) becomes \( -\frac{1}{2} \)
• A slope of \( -\frac{3}{4} \) becomes \( \frac{4}{3} \)

When two lines have slopes that are negative reciprocals of each other, they intersect at a right angle, forming a perfect 90-degree angle. Recognizing negative reciprocals allows us to quickly deduce the relationship between two lines and if they are perpendicular without needing to graph them.

Analytic Geometry

Analytic geometry, often called coordinate geometry, combines algebra and geometry to describe geometric figures using coordinates and equations. It allows us to compute various properties of shapes and lines directly using algebraic methods. A classic problem involves determining whether two lines are parallel, perpendicular, or neither.

• **Parallel Lines**: Have equal slopes \(( m_1 = m_2 )\)
• **Perpendicular Lines**: Slopes are negative reciprocals \(( m_1 = -\frac{1}{m_2} )\)
• **Neither**: No such relationship between slopes

By using the principles of analytic geometry, such as slope calculations and understanding of reciprocal relationships, we can solve complex geometric problems efficiently. It provides a powerful toolset for students and professionals to work with geometric problems using algebraic approaches, making it far easier to interpret and deduce relationships between geometric entities.