Problem 45 - Contemporary Precalculus

Question

Determine whether the line through \(P\) and \(Q\) is parallel or perpendicular to the line through \(P=(-3,1 / 3), Q=(1,-1)\) and \(R=(2,0), S=(4,-2 / 3)\)

Step-by-Step Solution

Answer

Answer: The lines are parallel.

Step 1: Find the slope of the line through points P and Q

To find the slope between two points, we use the following formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). In our case, \(x_1=-3, y_1=\frac{1}{3}, x_2=1,\) and \(y_2=-1\). Plugging these values into the formula: \(m_{PQ} = \frac{-1-\frac{1}{3}}{1-(-3)}\).

Step 2: Simplify the slope of the line through points P and Q

Now, we will simplify the slope: \(m_{PQ} = \frac{-\frac{4}{3}}{4} = -\frac{1}{3}\).

Step 3: Find the slope of the line through points R and S

Similarly to step 1, now we calculate the slope between points R and S using their coordinates: \(x_1=2, y_1=0, x_2=4,\) and \(y_2=-\frac{2}{3}\). \(m_{RS} = \frac{-\frac{2}{3}-0}{4-2}\).

Step 4: Simplify the slope of the line through points R and S

Now, we will simplify the slope: \(m_{RS} = \frac{-\frac{2}{3}}{2} = -\frac{1}{3}\).

Step 5: Compare the slopes

Since \(m_{PQ} = m_{RS} = -\frac{1}{3}\), the slopes are equal, which means the lines are parallel.

Step 6: Final Answer

The line through points P and Q is parallel to the line through points R and S.

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial in coordinate geometry, as it provides information about the direction and steepness of the line. The formula to calculate the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). The slope is a measure of how much the y-coordinate (vertical change) changes for a unit change in the x-coordinate (horizontal change).

A positive slope means the line rises from left to right, whereas a negative slope means the line falls from left to right. If the slope is zero, the line is horizontal, and if the slope is undefined (due to a division by zero), the line is vertical. In our exercise, we use this formula to find the slopes of lines through points \(P/Q\) and \(R/S\) respectively. Both slopes were found to be \( -\frac{1}{3} \), indicating both lines fall from left to right with the same steepness, hence they are parallel.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to describe geometric figures algebraically using coordinates and equations. It provides a link between geometry and algebra through graphs of lines and curves. This enables us to solve geometric problems by means of algebraic calculations.

For example, the positions of points \(P\), \(Q\), \(R\), and \(S\) are described using pairs of numbers corresponding to their location on the Cartesian plane. These coordinates represent positions along the horizontal (x-axis) and vertical (y-axis) dimensions. In the exercise, by computing the slopes of two lines through pairs of these points, we made use of the coordinate system to determine the relationship between the lines.

Linear Equations

Linear equations are algebraic equations in which the highest power of the variable is one. When graphed, they represent straight lines, hence they are vital in the study of lines in coordinate geometry. The most common form of a linear equation in two variables \( x \) and \( y \) is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, that is the point where the line crosses the y-axis.

Identifying whether lines are parallel or perpendicular is an essential skill in working with linear equations. As shown in the exercise, when two lines are parallel, they have the same slope. Conversely, if the product of the slopes of two lines is -1, the lines are perpendicular, showing an inverse relationship between the slopes of perpendicular lines. This knowledge can be applied to solve various geometric problems involving lines, angles, and shapes.