What’s the perpendicular slope of 4/3
The Correct Answer and Explanation is :
The perpendicular slope of a line is the negative reciprocal of the slope of the original line.
Given that the slope of the original line is ( \frac{4}{3} ), we can calculate the perpendicular slope as follows:
Step 1: Understanding the Concept of Perpendicular Slopes
In geometry, two lines are perpendicular if they intersect at a right angle (90 degrees). The slopes of two perpendicular lines are related in such a way that the product of their slopes is ( -1 ). This relationship is known as the negative reciprocal property.
If the slope of one line is ( m_1 ), the slope ( m_2 ) of any line perpendicular to it is given by:
[
m_2 = -\frac{1}{m_1}
]
Step 2: Applying the Formula to ( \frac{4}{3} )
Here, the slope of the given line is ( \frac{4}{3} ). To find the slope of the line perpendicular to it, we take the negative reciprocal of ( \frac{4}{3} ):
[
\text{Perpendicular slope} = -\frac{1}{\frac{4}{3}} = -\frac{3}{4}
]
Step 3: Conclusion
Therefore, the perpendicular slope to a line with a slope of ( \frac{4}{3} ) is ( -\frac{3}{4} ).
Why Does This Work?
The negative reciprocal relationship between perpendicular slopes ensures that the product of the slopes of two perpendicular lines is always ( -1 ). Mathematically, if you multiply the slopes ( \frac{4}{3} ) and ( -\frac{3}{4} ), you get:
[
\frac{4}{3} \times -\frac{3}{4} = -\frac{12}{12} = -1
]
This confirms that the lines are indeed perpendicular, as their slopes satisfy the condition for perpendicularity.