Question

Prove that if the slopes of two nonvertical lines are negative reciprocals of each other, then the lines are perpendicular.

Short answer

The proof is completed by using the definition of negative reciprocal slopes and the geometric property of perpendicular lines. If two lines have negative reciprocal slopes, then they are perpendicular.

Step-by-step solution

Define the slopes of the two lines

Let \(m_1\) and \(m_2\) represent the slopes of two nonvertical lines. According to the problem, \(m_1\) and \(m_2\) are negative reciprocals of each other, which means \(m_1 * m_2 = -1\).

Use the property of perpendicular lines

From the geometric properties, we know that two lines are perpendicular if the product of their slopes is -1. In other words, if two lines are perpendicular, their slopes \(m_1\) and \(m_2\) satisfy the equation \(m_1 * m_2 = -1\).

Use the definition to draw conclusion

So, if \(m_1\) and \(m_2\) are negative reciprocals of each other, they satisfy the equation \(m_1 * m_2 = -1\). Therefore, by definition, the two lines are perpendicular.