Which line is perpendicular to a line that has a slope of [tex]$\frac{1}{2}$[/tex]?
line AB
line CD
line FG
line HJ
Answer (2)
The problem requires finding a line perpendicular to a line with a slope of 2 1 .
Perpendicular lines have slopes whose product is -1.
Calculate the slope of the perpendicular line: m 2 = m 1 − 1 = 2 1 − 1 = − 2 .
The line with a slope of -2 is perpendicular to the given line. Without knowing the slopes of lines AB, CD, FG, and HJ, we cannot determine which one is perpendicular, but we know that the perpendicular line must have a slope of -2. − 2
Explanation
Problem Analysis We are given a line with a slope of 2 1 . We need to find which of the given lines (AB, CD, FG, HJ) is perpendicular to this line.
Perpendicular Lines Condition Two lines are perpendicular if the product of their slopes is -1. Let m 1 be the slope of the given line and m 2 be the slope of a line perpendicular to it. Then, m 1 ⋅ m 2 = − 1 .
Setting up the Equation We have m 1 = 2 1 . We need to find m 2 such that 2 1 ⋅ m 2 = − 1 .
Solving for the Perpendicular Slope To find m 2 , we can divide both sides of the equation by 2 1 : m 2 = 2 1 − 1 = − 1 ⋅ 1 2 = − 2 So, the slope of the perpendicular line is -2.
Finding the Correct Line Now we need to check which of the given lines (AB, CD, FG, HJ) has a slope of -2. Without knowing the slopes of lines AB, CD, FG, and HJ, we cannot determine which one is perpendicular. However, we know that the perpendicular line must have a slope of -2.
Examples
In architecture, ensuring walls are perpendicular to the ground is crucial for stability. If a wall needs to be built perpendicular to a slope, understanding the relationship between slopes of perpendicular lines is essential. For example, if the ground has a slope of 2 1 at a particular point, the supporting wall must be constructed with a slope of -2 to ensure it stands perfectly upright and provides the necessary structural support.
The line that is perpendicular to a line with a slope of 2 1 must have a slope of − 2 . Without knowing the slopes of lines AB, CD, FG, or HJ, we cannot determine which line is the correct answer. The critical point is that look for a line with slope − 2 .
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