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To find PN and QP when P is the centroid and QN=9, we use the centroid property that divides medians in a 2:1 ratio. Hence, PN is 6 and QP is 3.
The student is asking for help in determining the lengths of PN and QP, given that QN=9 and P is the centroid of the triangle ALMN.
In the context of geometry, the centroid of a triangle is the point where the triangle's three medians intersect. It also has the property that it divides each median into segments with a ratio of 2:1, with the longer part being nearest to the vertex.
Since QN represents one such segment (from the midpoint Q of the median to the vertex N), and P is the centroid, we know that PN is twice the length of QP.
We're given the total length of QN=9, which we can think of in terms of PN + QP.
Because QP is 1/3 of the whole segment (QN), and PN is 2/3, we can write:
QN = PN + QP
9 = (2/3 × QN) + (1/3 × QN)
Therefore:
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