This problem involves properties of a centroid in a triangle. The centroid $P$ divides each median into two segments such that the ratio of the longer segment (vertex to centroid) to the shorter segment (centroid to midpoint) is $2:1$. Let us calculate the missing values based on the given data:

Given:

• $PR = 12$ (median segment from $P$ to $R$),
• $PT = 8$ (median segment from $P$ to $T$),
• $AR = 9$ (entire median $AR$),
• $AY = 21$ (entire median $AY$).

To Find:

• $SP$,
• $TM$,
• $AT$,
• $PY$.

Step 1: Calculate $SP$ and $TM$

Using the $2:1$ centroid property:

• The total length of a median is divided into 3 equal parts. Thus, the part from $centroid \, P$ to the midpoint is $\frac{1}{3}$ of the entire median.
• The segment from the vertex to $P$ is $\frac{2}{3}$.

1. ( SP = \frac{1}{2} \times AY = \frac{1}{3} \times 11Let's calculate step outlines--Here's a clear breakdown:

Step-by-Step Solution:

1. Property of Centroid:
   • The centroid $P$ divides each median into a $2:1$ ratio.
   • The segment from the vertex to $P$ is $\frac{2}{3}$ of the median, while the segment from $P$ to the midpoint is $\frac{1}{3}$.

Step 2: Solve for the Missing Values:

1. Find $SP$:

   • The median $AY = 21$.
   • $SP = \frac{1}{3} \times AY = \frac{1}{3} \times 21 = 7$.

2. Find $PY$:

   • Using the $2:1$ ratio:
   • $PY = \frac{2}{3} \times AY = \frac{2}{3} \times 21 = 14$.

3. Find $TM$:

   • Median $AR = 9$.
   • $TM = \frac{1}{3} \times AR = \frac{1}{3} \times 9 = 3$.

4. Find $AT$:

   • Again using the $2:1$ ratio:
   • $AT = \frac{2}{3} \times AR = \frac{2}{3} \times 9 = 6$.

5. Check Given $PR$:

   • $PR = 12$, so this directly tells us the centroid-to-midpoint segment.

Final Results:

• $SP = 7$,
• $TM = 3$,
• $AT = 6$,
• $PY = 14$.

Let me know if you would like a detailed explanation of any step!