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 midsegment theorem

The midsegment of a triangle is parallel to the third side. Parallel lines have equal slopes.

Step2: Analyze each option

• Option A: If \( \overline{MN} \) is a midsegment, by the mid - segment theorem \( \overline{MN}\parallel\overline{AB} \). Since parallel lines have equal slopes, this option is correct.
• Option B: The slopes can be equal (when lines are parallel). So, this option is incorrect.
• Option C: The sign of the slope (positive or negative) depends on the orientation of the lines in the coordinate plane. A mid - segment can have a positive or negative slope depending on the triangle's position. So, this option is incorrect.
• Option D: Negative reciprocals of slopes imply perpendicular lines. But a mid - segment is parallel (not perpendicular) to the third side. So, this option is incorrect.

Answer:

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