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].
