Question

how can finding the slopes of mn and ab help in proving that mn is the midsegment of triangle abc?
a the slopes of mn and ab must be equal for mn to be a midsegment.
b the slopes of mn and ab cannot be equal for mn to be a midsegment.
c the slopes of mn and ab must be positive for mn to be a midsegment.
d the slopes of mn and ab must be negative reciprocals for mn to be a midsegment.

Explanation:

Step1: Recall the mid - segment theorem

The mid - segment of a triangle is parallel to the third side. If two lines are parallel, their slopes are equal.

Step2: Analyze each option

• Option A: If the slopes of \(MN\) and \(AB\) are equal, then \(MN\parallel AB\). By the mid - segment theorem (a line segment connecting the mid - points of two sides of a triangle and parallel to the third side is a mid - segment), this is a valid condition.
• Option B: The slopes do not need to be unequal for \(MN\) to be a mid - segment. As long as they are equal (for parallelism), it can be a mid - segment.
• Option C: The sign (positive or negative) of the slopes is not relevant for the mid - segment property. The key is the equality of slopes (for parallelism).
• Option D: Negative reciprocals of slopes imply perpendicular lines. But for a mid - segment, we need parallel lines (equal slopes), not perpendicular lines.

Answer:

A. The slopes of \(MN\) and \(AB\) must be equal for \(MN\) to be a midsegment.