How many lines can be drawn that contain exactly two of the six points when considering the midpoints of the sides of triangle ABC?

To determine the number of lines that can be drawn containing exactly two of the six points formed by the midpoints of the sides of triangle ABC, it's helpful to first visualize the scenario. A triangle has three sides, and the midpoint of each side creates a point. In triangle ABC:

• The midpoint of side AB

• The midpoint of side BC

• The midpoint of side CA

The three midpoints are distinct points on the triangle, and since we are considering the midpoints of all three sides, there are three midpoints. Additionally, if point A, B, and C are also included, this makes a total of six distinct points.

The task is to find out how many unique lines can be drawn that connect exactly two of these points. The key principle here involves combinations, specifically choosing 2 points from the set of 6.

The formula for combinations is given by:

[ C(n, k) = \frac{n!}{k!(n-k)!} ]

Where ( n ) is the total number of points, and ( k ) is the number of points to choose. Substituting in our values:

• ( n = 6 )

• ( k = 2 )

This gives