Answer :
The values are PN = 14 and QP = 28.
To find PN and QP, we need to understand the properties of a centroid in a triangle.
The centroid is the point of concurrency of the medians of a triangle. A median is a line segment drawn from a vertex to the midpoint of the opposite side. In other words, the centroid is the point where the medians intersect.
Given that point P is the centroid of triangle ALMN, we can deduce the following:
1. The median from vertex A passes through P and divides the opposite side LN into two equal segments. Let's call the point where the median intersects LN as point Q.
2. The median from vertex M also passes through P and divides the opposite side AL into two equal segments. Let's call the point where the median intersects AL as point N.
Now, let's find the lengths of PN and QP using the given information:
Given: ON = 42
Since the centroid divides the medians into a ratio of 2:1, we can determine that PN is one-third the length of ON. Similarly, QP is two-thirds the length of ON.
1. PN = (1/3) * ON = (1/3) * 42 = 14
2. QP = (2/3) * ON = (2/3) * 42 = 28
Therefore, PN = 14 and QP = 28.
