Final answer:
Using the properties of centroids in a triangle, we determine that the length PM is half of the full median PR, which is 9, making PM equal to 4.5.
Step-by-step explanation:
To find the length PM when K is the centroid of the triangle and given that NQ = 6, RK = 3, & PK = 4, we use properties of triangles and centroids.
The centroid of a triangle divides each median into two segments, with the segment connecting the centroid to the vertex being twice the length of the segment connecting the centroid to the midpoint of the opposite side. Since K is the centroid, PK:KR should be in the ratio 2:1 because PK is the segment connecting K to the vertex while KR is the segment connecting K to the midpoint of the opposite side.
Since RK = 3, therefore the full median PR must be 3 (RK) + 2x3 (PK) = 9. However, since PM is half of PR (because M is the midpoint), PM is 9/2 which equals to 4.5.