To find the lengths of the medians of the triangle $PQR$, we use the following approach:

Step 1: Calculate the midpoints of the sides of the triangle.

The median from a vertex of a triangle goes to the midpoint of the opposite side. So, we first need the midpoints of the sides opposite each vertex.

1. Midpoint of side $QR$ (opposite vertex $P$):

   $M_{QR} = \left( \frac{-6 + (-2)}{2}, \frac{9 + (-3)}{2} \right) = \left( \frac{-8}{2}, \frac{6}{2} \right) = (-4, 3)$

2. Midpoint of side $PR$ (opposite vertex $Q$):

   $M_{PR} = \left( \frac{10 + (-2)}{2}, \frac{1 + (-3)}{2} \right) = \left( \frac{8}{2}, \frac{-2}{2} \right) = (4, -1)$

3. Midpoint of side $PQ$ (opposite vertex $R$):

   $M_{PQ} = \left( \frac{10 + (-6)}{2}, \frac{1 + 9}{2} \right) = \left( \frac{4}{2}, \frac{10}{2} \right) = (2, 5)$

Step 2: Calculate the lengths of the medians.

We now compute the lengths of the medians using the distance formula between each vertex and the midpoint of the opposite side:

The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

1. Length of the median from $P$ to $M_{QR}$:

   $d_P = \sqrt{(-4 - 10)^2 + (3 - 1)^2} = \sqrt{(-14)^2 + (2)^2} = \sqrt{196 + 4} = \sqrt{200} = 10\sqrt{2} \approx 14.14$

2. Length of the median from $Q$ to $M_{PR}$:

   $d_Q = \sqrt{(4 - (-6))^2 + (-1 - 9)^2} = \sqrt{(4 + 6)^2 + (-1 - 9)^2} = \sqrt{10^2 + (-10)^2} = \sqrt{100 + 100} = \sqrt{200} = 10\sqrt{2} \approx 14.14$

3. Length of the median from $R$ to $M_{PQ}$:

   $d_R = \sqrt{(2 - (-2))^2 + (5 - (-3))^2} = \sqrt{(2 + 2)^2 + (5 + 3)^2} = \sqrt{4^2 + 8^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5} \approx 8.94$

Final Answer:

The lengths of the medians of the triangle are approximately:

1. $10\sqrt{2} \approx 14.14$ units (from $P$)
2. $10\sqrt{2} \approx 14.14$ units (from $Q$)
3. $4\sqrt{5} \approx 8.94$ units (from $R$)

Would you like more details on any step?

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Here are five related questions you can explore:

1. How are medians used in finding the centroid of a triangle?
2. What is the formula for finding the centroid of a triangle given its vertices?
3. How can you prove that the centroid divides each median in a 2:1 ratio?
4. How are the lengths of medians related to the sides of the triangle?
5. Can the medians be used to find the area of the triangle?

Tip: The centroid of a triangle is always located at the point where the three medians intersect, and it divides each median into two segments in the ratio 2:1.