Question

in exercises 1–4, point ( p ) is the centroid of ( \triangle lmn ). find ( pn ) and ( qp ).

1. ( qn = 9 )
2. ( qn = 21 )
3. ( qn = 30 )
4. ( qn = 42 )

in exercises 5–8, point ( d ) is the centroid of ( \triangle abc ). find ( cd ) and ( ce ).

Explanation:

Step1: Recall centroid segment ratio

The centroid of a triangle divides a median into a ratio of $2:1$, where the longer segment is from the vertex to the centroid. So $QP = \frac{1}{3}QN$ and $PN = \frac{2}{3}QN$.
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For 1. $QN=9$

Step1: Calculate $QP$

$QP = \frac{1}{3} \times 9$

Step2: Calculate $PN$

$PN = \frac{2}{3} \times 9$

For 2. $QN=21$

Step1: Calculate $QP$

$QP = \frac{1}{3} \times 21$

Step2: Calculate $PN$

$PN = \frac{2}{3} \times 21$

For 3. $QN=30$

Step1: Calculate $QP$

$QP = \frac{1}{3} \times 30$

Step2: Calculate $PN$

$PN = \frac{2}{3} \times 30$

For 4. $QN=42$

Step1: Calculate $QP$

$QP = \frac{1}{3} \times 42$

Step2: Calculate $PN$

$PN = \frac{2}{3} \times 42$

Answer:

1. $QP=3$, $PN=6$
2. $QP=7$, $PN=14$
3. $QP=10$, $PN=20$
4. $QP=14$, $PN=28$