PN is 34 units long and QN is 17 units long.
To find the lengths of PN and QN, given that point P is the centroid of triangle LMN and QN is 17 units long, we can use the properties of a centroid in a triangle.
In a triangle with centroid as P, the centroid divides each median into segments where the ratio of the smaller segment to the larger segment is 2:1. This means that the segment from the centroid to the vertex is twice as long as the segment from the centroid to the midpoint of the opposite side.
Let's denote:
PN as x
QN as 17 units
Given that P is the centroid, PN to QN ratio is 2:1. So, we can set up an equation using this ratio:
QN/ PN = 1/ 2
Substitute the values:
17/ x = 1/ 2
Cross-multiply:
2×17=x×1
x=34
Therefore, PN is 34 units long and QN is 17 units long.