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Which line is perpendicular to a line that has a slope of [tex]-\frac{5}{6}[/tex]?

A. line JK
B. line LM
C. line NO
D. line PQ


Sagot :

To determine which line is perpendicular to a line with a given slope, we need to use the properties of perpendicular lines in coordinate geometry.

1. Understand the Slope of Perpendicular Lines:

In coordinate geometry, when two lines are perpendicular to each other, the product of their slopes is -1. That is, if one line has a slope [tex]\( m_1 \)[/tex], then the slope [tex]\( m_2 \)[/tex] of the line perpendicular to it can be found using:
[tex]\[ m_1 \times m_2 = -1 \][/tex]

2. Identify the Given Slope:

The slope of the given line is [tex]\( m_1 = -\frac{5}{6} \)[/tex].

3. Find the Slope of the Perpendicular Line:

Let [tex]\( m_2 \)[/tex] be the slope of the line that is perpendicular to the given line. We know:
[tex]\[ -\frac{5}{6} \times m_2 = -1 \][/tex]

To isolate [tex]\( m_2 \)[/tex], divide both sides of the equation by [tex]\( -\frac{5}{6} \)[/tex]:
[tex]\[ m_2 = \frac{-1}{-\frac{5}{6}} = \frac{-1 \times 6}{-5} = \frac{6}{5} \][/tex]

4. Simplify the Slope:

Simplifying the fraction, we get:
[tex]\[ m_2 = 1.2 \][/tex]

Thus, the slope of any line that is perpendicular to the given line with a slope of [tex]\( -\frac{5}{6} \)[/tex] is [tex]\( 1.2 \)[/tex].

To conclude, a line that is perpendicular to a line with a slope of [tex]\( -\frac{5}{6} \)[/tex] has a slope of [tex]\( 1.2 \)[/tex]. Since we are not provided with specific slopes of lines JK, LM, NO, or [tex]\( PQ \)[/tex], the general answer is: Any line with a slope of [tex]\( 1.2 \)[/tex] will be perpendicular to the line with a slope of [tex]\( -\frac{5}{6} \)[/tex].