Question

in $\triangle cde$, $u$ is the centroid, $uk = 12$, $em = 21$, and $ud = 9$. find $mu$.

Explanation:

Step1: Recall centroid segment ratio

The centroid of a triangle divides a median into a ratio of $2:1$, where the segment from the vertex to the centroid is twice the segment from the centroid to the midpoint. So for median $EM$, $EU:MU = 2:1$.

Step2: Define variables for segments

Let $MU = x$. Then $EU = 2x$.

Step3: Set up total length equation

The total length of median $EM$ is $EU + MU = 21$. Substitute the variables:
$2x + x = 21$

Step4: Solve for $x$

Combine like terms: $3x = 21$
Solve for $x$: $x = \frac{21}{3} = 7$

Answer:

7