A cube has 8 vertices. For each pair of distinct vertices, we connect them up with a line segment. There are
(8)
(2) =28
such line segments. For each of these 28 line segments, we mark the midpoint. How many distinct points have been marked as the midpoints?
19
To determine how many distinct points have been marked as midpoints, we need to calculate the number of unique midpoints among the 28 line segments.
To find the number of unique midpoints, we can start by considering the number of ways we can choose two vertices out of the eight. This can be calculated using the binomial coefficient formula:
C(n, k) = (n!)/(k!(n-k)!)
In this case, we want to find C(8, 2) which means choosing 2 vertices out of 8. Plugging these values into the formula, we get:
C(8, 2) = (8!)/(2!(8-2)!)
= (8!)/(2!6!)
= (8 * 7 * 6!)/(2!6!)
= (8 * 7)/2!
= 8 * 7/2
= 56/2
= 28
So we have 28 distinct line segments connecting pairs of distinct vertices.
Now, for each of the 28 line segments, we mark the midpoint. Since we have already established that there are 28 line segments, the number of midpoints will also be 28.
Therefore, the answer is that there are 28 distinct points marked as midpoints.