If a line has a slope of 3/4, what is the slope of the line perpendicular to the line?
The Correct Answer and Explanation is:
The slope of the line perpendicular to a line with a slope of ( \frac{3}{4} ) is ( -\frac{4}{3} ).
Explanation:
In geometry, the slope of a line represents how steep the line is. Slope is often denoted by the letter ( m ) and is calculated as the “rise over run,” or the change in ( y )-coordinates divided by the change in ( x )-coordinates between two points on the line.
For two lines to be perpendicular, their slopes must be negative reciprocals of each other. This means that when you multiply the slopes of two perpendicular lines, the result should be ( -1 ). Mathematically, if the slope of one line is ( m_1 ), and the slope of the perpendicular line is ( m_2 ), then the relationship is:
[
m_1 \times m_2 = -1
]
In this problem, the slope of the first line, ( m_1 ), is ( \frac{3}{4} ). To find the slope of the line perpendicular to it, ( m_2 ), we use the negative reciprocal of ( \frac{3}{4} ). The reciprocal of ( \frac{3}{4} ) is ( \frac{4}{3} ), and the negative reciprocal would be ( -\frac{4}{3} ).
To verify this, let’s multiply the two slopes:
[
\frac{3}{4} \times -\frac{4}{3} = -\frac{12}{12} = -1
]
Since the product of the slopes is ( -1 ), the lines are indeed perpendicular.
In summary, the slope of a line perpendicular to one with a slope of ( \frac{3}{4} ) is ( -\frac{4}{3} ). This concept is useful in coordinate geometry when determining relationships between lines, particularly in problems involving right angles, constructing perpendicular bisectors, or analyzing geometric shapes.