Problem 30 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the lines through each pair of points are perpendicular. $$(-1,-6)\( and \)(2,6) ;(-8,-1)\( and \)(4,2)$$

Step-by-Step Solution

Answer

No, the lines passing through the pairs of points provided aren't perpendicular.

Step 1: Identify the Pairs of Points

The pairs of points provided are (-1,-6) and (2,6), and (-8,-1) and (4,2). These sets of points will determine two different lines.

Step 2: Calculate the Slopes

The slope of a line passing through points (x1,y1) and (x2,y2) is calculated as (y2 - y1) / (x2 - x1). The first line passes through the points (-1,-6) and (2,6), so its slope (m1) can be found as: \(m_1 = \frac{6 - (-6)}{2 - (-1)} = 2\). The second line passes through the points (-8,-1) and (4,2), so its slope (m2) can be calculated as: \(m_2 = \frac{2 - (-1)}{4 - (-8)} = 1/3\).

Step 3: Confirm Perpendicularity

Two lines are perpendicular if the product of their slopes equals -1. Here, \(m_1 * m_2 = 2 * 1/3 = 2/3\), which is not equal to -1. Therefore, the lines aren't perpendicular.

Key Concepts

Calculating Slope

The slope of a line essentially measures its steepness. This is an important geometric concept, particularly when determining relationships between lines, such as parallelism or perpendicularity. To calculate the slope of a line passing through two given points, denoted as

• \((x_1, y_1)\)
• \((x_2, y_2)\)

use the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\].
For example, to find the slope between the points \((-1, -6)\) and \((2, 6)\), substitute into the slope formula:
This yields: \[m = \frac{6 - (-6)}{2 - (-1)} = \frac{12}{3} = 4\].
Alongside, calculate the slope for another set of points, say \((-8, -1)\) and \((4, 2)\):\[m = \frac{2 - (-1)}{4 - (-8)} = \frac{3}{12} = \frac{1}{4}\].
Once the slopes of the lines are obtained, they play a crucial role in investigating if the lines are perpendicular.

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line so that it is clear what the slope and y-intercept are. It is expressed as
\[ y = mx + b \],
where

• \(m\) is the slope of the line
• \(b\) is the y-intercept, or the point where the line crosses the y-axis.

This form makes it straightforward to understand the line's angle of inclination (positive/negative slope) and its height at the y-intercept.
For example, using the points \((-1, -6)\) and \((2, 6)\) and a slope \(m = 4\), you'd insert a point such as \((x_1, y_1) = (-1, -6)\) into the equation \[y = 4x + b \] to find \(b\). Solving gives \[-6 = 4(-1) + b \rightarrow -6 = -4 + b \rightarrow b = -2 \].
Thus, the equation for this line is \[ y = 4x - 2 \].
Slope-intercept form is a nifty tool for quickly sketching lines and understanding their geometric properties.

Geometry

In geometry, understanding relationships between lines is crucial, especially when they intersect or form specific angles. **Perpendicularity**, a key geometric concept, describes when two lines intersect at a right angle (90 degrees).
To verify whether two lines are perpendicular, look at the relationship between their slopes. Lines are perpendicular if the product of their slopes is equal to \(-1\).
From this problem, for instance, let's consider the slopes

• \(m_1 = 2\)
• \(m_2 = \frac{1}{3}\)

The product of these slopes is \(2 \times \frac{1}{3} = \frac{2}{3}\), which isn't \(-1\).
Thus, these lines are not perpendicular. Knowing how to determine when lines are perpendicular is significant in countless geometric applications and real-world problems, such as constructing buildings or roadways.