Problem 50 - Contemporary Precalculus

Question

Determine whether the lines whose equations are given are parallel, perpendicular, or neither. Do the points \((-4,6),(-1,12),\) and (-7,0) all lie on the same straight line? [Hint: Use slopes.]

Step-by-Step Solution

Answer

Answer: Yes, the points lie on the same straight line. Since they are on the same line, the question about the lines being parallel, perpendicular, or neither does not apply in this case.

Step 1: Calculate the slope between each pair of points

The formula for calculating the slope of a line between two points is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). We have three pairs of points: \((-4,6)\) and \((-1,12)\), \((-1,12)\) and \((-7,0)\), and \((-4,6)\) and \((-7,0)\). Let's calculate the slopes between them. For \((-4,6)\) and \((-1,12)\): \(m_1 = \frac{12 - 6}{-1 - (-4)} = \frac{6}{3} = 2\) For \((-1,12)\) and \((-7,0)\): \(m_2 = \frac{0 - 12}{-7 - (-1)} = \frac{-12}{-6} = 2\) For \((-4,6)\) and \((-7,0)\): \(m_3 = \frac{0 - 6}{-7 - (-4)} = \frac{-6}{-3} = 2\)

Step 2: Check if all slopes are the same

Now we check if the calculated slopes \(m_1\), \(m_2\), and \(m_3\) are the same: \(m_1 = 2\) \(m_2 = 2\) \(m_3 = 2\) Since all the slopes are the same, it means that all the points lie on the same straight line.

Key Concepts

Slope Calculation

Understanding how to calculate the slope is fundamental in analytical geometry. The slope of a line is essentially a measure of its steepness. By using two points on the line, we can compute the slope using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula signifies the change in the y-values divided by the change in the x-values. Think of it as how much y increases or decreases when you move along the x-axis. Slope can be positive, negative, zero, or undefined, which reflects different orientations of a line:

• Positive Slope: The line rises as it moves from left to right.
• Negative Slope: The line falls as it moves from left to right.
• Zero Slope: The line is horizontal and doesn’t rise or fall.
• Undefined Slope: The line is vertical and doesn’t move horizontally.

Calculating slopes accurately helps to determine relationships, such as parallelism or perpendicularity, between lines.

Parallel and Perpendicular Lines

Let's dive deeper into how slopes help identify parallel and perpendicular lines. Two lines are parallel if and only if their slopes are equal. This means they rise or fall at the same rate, and thus, they never intersect. For example, in the coordinate plane, if lines have slopes \( m_1 = 2 \) and \( m_2 = 2 \), they are parallel.In contrast, perpendicular lines have slopes that are negative reciprocals of each other. This means if one line's slope is \( m_1 \), the perpendicular line's slope \( m_2 \) would be \( -\frac{1}{m_1} \). For instance, if the slope of one line is \( 2 \), a line perpendicular to it would have a slope of \( -\frac{1}{2} \). This perpendicular relationship makes a 'T' shape at their intersection, forming a right angle.

Collinear Points

Collinear points lie on the same straight line, and finding them involves using slopes effectively. If several points are collinear, the slopes between each pair of points should be identical, indicating they form part of the same line.Using the given example, when checking if the three points \((-4,6)\), \((-1,12)\), and \((-7,0)\) are collinear, we calculate slopes between each pair:

• From \((-4,6)\) to \((-1,12)\), the slope \( m_1 = 2 \).
• From \((-1,12)\) to \((-7,0)\), the slope \( m_2 = 2 \).
• From \((-4,6)\) to \((-7,0)\), the slope \( m_3 = 2 \).

Since \( m_1 = m_2 = m_3 = 2 \), these points align on the same line, confirming they are collinear. Recognizing collinear points is crucial in understanding the geometric relationships among points in a plane.