Final answer:
The correct relationship between the slopes of lines HN and PN, which are perpendicular to each other, is that the slope of one is the negative reciprocal of the other, indicating option (a) mHN = -1/mPN is correct.
Step-by-step explanation:
To prove that the slopes of lines HN and PN are opposite reciprocals, we must first understand the relationship between slopes of perpendicular lines. If two lines are perpendicular, their slopes are opposite reciprocals of each other. This concept can be derived from the slope-intercept form of a linear equation, which is written as y = mx + b, where m represents the slope.
Based on the student's question, it seems they need to determine which option correctly identifies the relationship between mHN and mPN. If mHN and mPN represent the slopes of line HN and PN respectively, and these lines are known to be perpendicular, then the correct answer would be:
This signifies that the slope of line HN is the negative reciprocal of the slope of line PN. This relationship is valid because perpendicular lines in a plane have slopes that multiply to -1. For example, if line HN had a slope of 2, then line PN would need to have a slope of -1/2 to be perpendicular to line HN.