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The problem is in Gujarati and can be translated to:

**"If A(5, -1), B(-3, -2), and C(-1, 8) are the vertices of triangle ABC, find the length of the median from vertex A."**

To solve this, we need to determine the length of the median from vertex A to the opposite side BC. The median from vertex A is the line segment joining A to the midpoint of the line segment BC. Let’s calculate this step by step:

### Step 1: Find the midpoint of BC
The coordinates of the midpoint \( M \) of segment \( BC \) can be found using the midpoint formula:

\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of points B and C.

For points B(-3, -2) and C(-1, 8):
\[
M = \left( \frac{-3 + (-1)}{2}, \frac{-2 + 8}{2} \right)
\]
\[
M = \left( \frac{-4}{2}, \frac{6}{2} \right)
\]
\[
M = (-2, 3)
\]

### Step 2: Find the length of the median AM
The length of the median from point A to the midpoint M is given by the distance formula:

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

where \((x_1, y_1) = (5, -1)\) (coordinates of A) and \((x_2, y_2) = (-2, 3)\) (coordinates of M).

\[
d = \sqrt{(-2 - 5)^2 + (3 - (-1))^2}
\]
\[
d = \sqrt{(-7)^2 + (4)^2}
\]
\[
d = \sqrt{49 + 16}
\]
\[
d = \sqrt{65}
\]

Thus, the length of the median from vertex A is \(\sqrt{65}\).

Would you like more details on this solution, or do you have any other questions?

Here are some related questions you might find interesting:
1. How do you calculate the centroid of a triangle given its vertices?
2. What is the formula for calculating the distance between two points?
3. How would you determine if a given triangle is isosceles using its vertices?
4. How do you find the equation of a median in a triangle?
5. What are the properties of medians in different types of triangles?

**Tip:** Remember, the midpoint formula and distance formula are fundamental tools in coordinate geometry, useful for analyzing shapes and distances in the plane.

If A(5, -1), B(-3, -2), and C(-1, 8) are the vertices of triangle ABC, find the length of the median from vertex A.

Learn how to find the length of the median from vertex A in triangle ABC with given vertices A(5, -1), B(-3, -2), and C(-1, 8). This problem involves the midpoint and distance formulas, covering important concepts in coordinate geometry. Suitable for students in Grades 9-12.

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