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Answer :
To determine which line is perpendicular to a line with a slope of [tex]\(-\frac{5}{6}\)[/tex], we need to consider the concept of perpendicular slopes. Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex].
Here’s a step-by-step explanation:
1. Understand the Relationship: If the slope of one line is [tex]\(m_1\)[/tex], and it is perpendicular to another line with slope [tex]\(m_2\)[/tex], then:
[tex]\[
m_1 \times m_2 = -1
\][/tex]
2. Given Slope: The slope of the given line is [tex]\(-\frac{5}{6}\)[/tex].
3. Find the Perpendicular Slope:
- Let [tex]\(m_2\)[/tex] be the slope of the line that is perpendicular to our line.
- We set up the equation:
[tex]\[
-\frac{5}{6} \times m_2 = -1
\][/tex]
4. Solve for [tex]\(m_2\)[/tex]:
- To find [tex]\(m_2\)[/tex], divide both sides of the equation by [tex]\(-\frac{5}{6}\)[/tex]:
[tex]\[
m_2 = \frac{1}{-\frac{5}{6}}
\][/tex]
- Simplify the fraction:
[tex]\[
m_2 = \frac{-6}{5}
\][/tex]
- Therefore, the slope of the perpendicular line is [tex]\(\frac{6}{5}\)[/tex].
5. Conclusion: Any line with a slope of [tex]\(\frac{6}{5}\)[/tex] will be perpendicular to the line with a slope of [tex]\(-\frac{5}{6}\)[/tex].
So, the correct answer is that the slope of the line that is perpendicular to the given line should be [tex]\(\frac{6}{5}\)[/tex] or numerically [tex]\(\approx 1.2\)[/tex]. This would identify which specific line (out of JK, LM, NO, PQ) is perpendicular if their slopes were provided.
Here’s a step-by-step explanation:
1. Understand the Relationship: If the slope of one line is [tex]\(m_1\)[/tex], and it is perpendicular to another line with slope [tex]\(m_2\)[/tex], then:
[tex]\[
m_1 \times m_2 = -1
\][/tex]
2. Given Slope: The slope of the given line is [tex]\(-\frac{5}{6}\)[/tex].
3. Find the Perpendicular Slope:
- Let [tex]\(m_2\)[/tex] be the slope of the line that is perpendicular to our line.
- We set up the equation:
[tex]\[
-\frac{5}{6} \times m_2 = -1
\][/tex]
4. Solve for [tex]\(m_2\)[/tex]:
- To find [tex]\(m_2\)[/tex], divide both sides of the equation by [tex]\(-\frac{5}{6}\)[/tex]:
[tex]\[
m_2 = \frac{1}{-\frac{5}{6}}
\][/tex]
- Simplify the fraction:
[tex]\[
m_2 = \frac{-6}{5}
\][/tex]
- Therefore, the slope of the perpendicular line is [tex]\(\frac{6}{5}\)[/tex].
5. Conclusion: Any line with a slope of [tex]\(\frac{6}{5}\)[/tex] will be perpendicular to the line with a slope of [tex]\(-\frac{5}{6}\)[/tex].
So, the correct answer is that the slope of the line that is perpendicular to the given line should be [tex]\(\frac{6}{5}\)[/tex] or numerically [tex]\(\approx 1.2\)[/tex]. This would identify which specific line (out of JK, LM, NO, PQ) is perpendicular if their slopes were provided.
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