Question

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

1. ( de = 5 )
2. ( de = 11 )
3. ( de = 9 )
4. ( de = 15 )

Explanation:

Step1: Recall centroid segment ratio

The centroid divides a median into a ratio of $2:1$, where the segment from vertex to centroid is twice the segment from centroid to the midpoint. So, if $DE$ is the shorter segment, the length of the entire median $CE = CD + DE$, and $CD = 2 \times DE$.
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For Exercise 5: $DE=5$

Step1: Calculate $CD$

$CD = 2 \times DE = 2 \times 5 = 10$

Step2: Calculate $CE$

$CE = CD + DE = 10 + 5 = 15$

For Exercise 6: $DE=11$

Step1: Calculate $CD$

$CD = 2 \times DE = 2 \times 11 = 22$

Step2: Calculate $CE$

$CE = CD + DE = 22 + 11 = 33$

For Exercise 7: $DE=9$

Step1: Calculate $CD$

$CD = 2 \times DE = 2 \times 9 = 18$

Step2: Calculate $CE$

$CE = CD + DE = 18 + 9 = 27$

For Exercise 8: $DE=15$

Step1: Calculate $CD$

$CD = 2 \times DE = 2 \times 15 = 30$

Step2: Calculate $CE$

$CE = CD + DE = 30 + 15 = 45$

Answer:

1. $CD=10$, $CE=15$
2. $CD=22$, $CE=33$
3. $CD=18$, $CE=27$
4. $CD=30$, $CE=45$