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Question

Lines l and n intersect at point upper A. Line l is increasing from left to right and line n is decreasing from left to right. The bottom angle at the point upper A has a right angle symbol.

Given that lines l
and n
are perpendicular, which of the following is part of the proof that l
and n
have slopes that are opposite reciprocals?

Answer 1

To prove that lines \( l \) and \( n \) have slopes that are opposite reciprocals, you can start from the fact that they are perpendicular to each other. Here’s a part of the proof you would typically include:

1. Definition of Slopes:

   • Let the slope of line \( l \) be denoted as \( m_l \) and the slope of line \( n \) be denoted as \( m_n \).

2. Perpendicular Lines:

   • According to the property of perpendicular lines, if two lines are perpendicular, then the product of their slopes equals -1. This can be expressed mathematically as: \[ m_l \cdot m_n = -1 \]

3. Implication:

   • From the equation \( m_l \cdot m_n = -1 \), we can rearrange this to show that: \[ m_n = -\frac{1}{m_l} \]
   • This shows that the slopes \( m_l \) and \( m_n \) are opposite reciprocals of each other.

This reasoning shows that when two lines are perpendicular, their slopes relate as stated, demonstrating that \( l \) and \( n \) have slopes that are opposite reciprocals.