find the length of the three medians of the triangle PQR whose vertices are P=(10,1), Q=(-6,9) and R=(-2, -3).

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.

Find the length of the three medians of the triangle PQR whose vertices are P=(10,1), Q=(-6,9) and R=(-2, -3).

Discover how to find the lengths of the medians of triangle PQR with a detailed solution using the midpoint and distance formulas. Learn key geometric concepts like medians, centroids, and the distance formula, suitable for high school students in Grades 9-12.

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