Final answer:
To find QN and QP when Point P is the centroid of triangle LMN and PN is 38 units, calculate QN as 1.5 times PN (57 units), and QP as one-third of PN (approximately 12.67 units).
Step-by-step explanation:
The student has asked how to find the lengths QN and QP given that Point P is the centroid of triangle LMN and PN equals 38 units. In a triangle, the centroid is the point where the medians intersect, and it is also known as the center of mass or barycenter of the triangle. One important property of the centroid is that it divides each median into segments with a ratio of 2:1, with the larger segment being closer to the vertex opposite the side the median is drawn to.
Since P is the centroid and PN is a segment of the median coming from vertex N, the full length of the median would be one and a half times PN. Therefore, QN equals 1.5 times PN. Since PN is 38 units, QN would be 38 units times 1.5, which equals 57 units.
To find the length of QP, which is the shorter segment of the median from Q to P, we use the fact that QP is one-third of the whole median (since P divides the median in a 2:1 ratio). Therefore, QP is 38 units divided by 3, which equals approximately 12.67 units.