6. in △cde, u is the centroid, uk = 12, em = 21, and ud = 9. find each measure. 1. cu 2. mu 3. eu 4. jd

Question

Accepted Answer

### 1. Finding $ CU $
# Explanation:
## Step1: Recall centroid property
The centroid of a triangle divides each median into a ratio of $ 2:1 $, with the longer segment being closer to the vertex. For median $ CK $ (where $ K $ is the midpoint of $ DE $), $ CU:UK = 2:1 $.
## Step2: Calculate $ CU $
Given $ UK = 12 $, let $ CU = 2x $ and $ UK = x $. So $ x = 12 $, then $ CU = 2\times12 = 24 $.
# Answer:
$ 24 $

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### 2. Finding $ MU $
# Explanation:
## Step1: Recall centroid property for median $ EM $
The centroid $ U $ divides median $ EM $ such that $ EU:MU = 2:1 $, and $ EM = EU + MU $. Given $ EM = 21 $, let $ MU = x $, then $ EU = 2x $, so $ 2x + x = 21 $.
## Step2: Solve for $ x $
$ 3x = 21 $, so $ x = \frac{21}{3} = 7 $. Thus, $ MU = 7 $.
# Answer:
$ 7 $

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### 3. Finding $ EU $
# Explanation:
## Step1: Use centroid property on median $ EM $
As the centroid divides the median $ EM $ in ratio $ 2:1 $ ( $ EU:MU = 2:1 $ ) and $ EM = 21 $. Let $ EU = 2x $ and $ MU = x $, so $ 2x + x = 21 $ (from Step 2 of finding $ MU $).
## Step2: Calculate $ EU $
We know $ x = 7 $ (from $ MU $ calculation), so $ EU = 2\times7 = 14 $.
# Answer:
$ 14 $

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### 4. Finding $ JD $
# Explanation:
## Step1: Recall centroid property for median $ JD $ (where $ J $ is the midpoint of $ CE $)
The centroid $ U $ divides median $ JD $ such that $ JU:UD = 2:1 $. Given $ UD = 9 $, let $ JU = 2x $ and $ UD = x $. So $ x = 9 $, then $ JU = 2\times9 = 18 $.
## Step2: Calculate $ JD $
$ JD = JU + UD = 18 + 9 = 27 $.
# Answer:
$ 27 $

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QUESTION IMAGE

Question

Explanation:

1. Finding \( CU \)

Step1: Recall centroid property

Step2: Calculate \( CU \)

Step1: Recall centroid property for median \( EM \)

Step2: Solve for \( x \)

Step1: Use centroid property on median \( EM \)

Step2: Calculate \( EU \)

Answer:

2. Finding \( MU \)