A line has a slope of 4. What is the slope of any line perpendicular to this line? A −4 B −14 C 14 D undefined
The Correct Answer and Explanation is :
The correct answer is: (-\frac{1}{4})
To find the slope of a line that is perpendicular to another line, we use the property that the product of the slopes of two perpendicular lines is (-1). This means if one line has a slope (m_1) and another line has a slope (m_2), then:
[ m_1 \times m_2 = -1 ]
Given that the slope of the first line is (m_1 = 4), we can set up the equation to find (m_2), the slope of the line perpendicular to it:
[ 4 \times m_2 = -1 ]
To isolate (m_2), we divide both sides of the equation by (4):
[ m_2 = \frac{-1}{4} ]
Thus, the slope of the line that is perpendicular to the original line with a slope of (4) is (-\frac{1}{4}).
However, let’s analyze the provided answer choices:
- A: (-4)
- B: (-14)
- C: (14)
- D: undefined
None of these options match (-\frac{1}{4}). It is possible that there was a misunderstanding in the question or the answer choices provided. The correct slope for any line perpendicular to a line with a slope of (4) is indeed (-\frac{1}{4}).
Summary
In summary, the slope of a line that is perpendicular to another line with a slope of (4) is (-\frac{1}{4}). The concept of perpendicular slopes is fundamental in geometry and algebra. Remembering that the product of the slopes must equal (-1) allows you to quickly calculate perpendicular slopes, making it a useful tool in various mathematical applications, including analytic geometry, calculus, and more advanced topics in mathematics. Understanding this relationship deepens comprehension of the behavior of lines in a coordinate plane.