Question: User: M
T
E
L
32
P
18.6
N
C
Additional Information
CT=45.6
Graph description: top-left triangle/line network with points M (top-left), T (top-center), E (top-right), L (left-middle), C (bottom-left), P (center), and N (lower-right of center); thick outer segments M-C and C-E and top segment M-E, with interior segments M-P-N and C-P-T and L-P-E. Known labels: MP=32, LP=18.6, CT=45.6. position_hint: top-leftGiven that point P is the centroid of \u25B2MEC, what is the length of LE?
37.2
55.8
42.3
47.8
1. **Problem Statement:** Given triangle $\triangle MEC$ with centroid $P$, and lengths $MP=32$, $LP=18.6$, and $CT=45.6$, find the length of segment $LE$.
2. **Key Concept:** The centroid of a triangle divides each median into a ratio of $2:1$, with the longer segment adjacent to the vertex.
3. **Identify Medians:** Since $P$ is the centroid of $\triangle MEC$, $P$ lies on medians from vertices $M$, $E$, and $C$.
4. **Given Lengths:** $MP=32$ is two-thirds of median $ME$ (from $M$ to midpoint of $EC$), so the full median $ME = \frac{3}{2} \times 32 = 48$.
5. **Similarly, $LP=18.6$ is part of median $LE$. Since $L$ is on side $MC$, and $P$ lies on median $LE$, $LP$ is two-thirds of $LE$.
6. **Calculate $LE$:**
$$LE = \frac{3}{2} \times 18.6 = 27.9$$
7. **Given $CT=45.6$, which is a segment related to the triangle but not directly needed for $LE$ calculation.**
8. **Check options:** None match $27.9$ exactly, so re-examine the problem.
9. **Note:** The problem asks for length $LE$, and $LP=18.6$ is given. Since $P$ is centroid, $LP$ is two-thirds of $LE$, so:
$$LE = \frac{3}{2} \times 18.6 = 27.9$$
10. **However, the options are 37.2, 55.8, 42.3, 47.8.**
11. **Possibility:** $LP=18.6$ might be one-third of $LE$ instead, so:
$$LE = 3 \times 18.6 = 55.8$$
12. **This matches one of the options.**
13. **Final answer:** $LE = 55.8$.
**Answer:** $\boxed{55.8}$