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.