Problem 34 - Precalculus Essentials

Question

If two lines are perpendicular, describe the relationship between their slopes.

Step-by-Step Solution

Answer

The relationship between the slopes of two perpendicular lines is that their product is -1. Each slope is the negative reciprocal of the other.

Step 1: Understanding the concept of Perpendicular lines

In simple geometric terms, two lines are said to be perpendicular if they intersect each other at a right angle (90 degrees). So if we have two lines, line 1 and line 2, and they meet at a point forming a right angle, then line 1 is perpendicular to line 2 and vice versa.

Step 2: Understanding the concept of Slope

The slope of a line is a measure of its steepness or the 'tilt'. Mathematically, it's calculated as the ratio of the vertical change (y-axis) to the horizontal change (x-axis), and is usually represented as \( m \) in equations of lines. The slope of a line is calculated as \( m = \frac{\Delta y}{\Delta x} \), where \( \Delta y \) is the change in y and \( \Delta x \) is the change in x.

Step 3: Relationship between the slopes of Perpendicular lines

This is the key step in this exercise. The relationship between the slopes of two perpendicular lines is such that the product of their slopes is -1. So, if the slope of line 1 is \( m_1 \) and the slope of line 2 is \( m_2 \), and assuming the lines are perpendicular, then \( m_1 × m_2 = -1 \). This implies each slope is the negative reciprocal of the other. For example, if the slope of line 1 is 2, the slope of line 2 which is perpendicular to line 1 would be \( \frac{-1}{2} \).