(1,-1) (0,4) and (-5,3) are the vertices of triangle .Check whether it is a
scalene triangle, isosceles triangle or an equilateral triangle.Also find the
length of the median joining the vertex(1,-1) and mid point of opposite
side. 

To determine whether the given triangle is scalene, isosceles, or equilateral, we need to calculate the lengths of all three sides of the triangle. 

The vertices of the triangle are:
\[
A(1, -1), \quad B(0, 4), \quad C(-5, 3)
\]

### Step 1: Calculate the lengths of the sides of the triangle

We use the distance formula to find the lengths of the sides:

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

#### Length of side \(AB\):

\[
AB = \sqrt{(0 - 1)^2 + (4 - (-1))^2}
\]

\[
AB = \sqrt{(-1)^2 + (5)^2}
\]

\[
AB = \sqrt{1 + 25} = \sqrt{26}
\]

#### Length of side \(BC\):

\[
BC = \sqrt{(-5 - 0)^2 + (3 - 4)^2}
\]

\[
BC = \sqrt{(-5)^2 + (-1)^2}
\]

\[
BC = \sqrt{25 + 1} = \sqrt{26}
\]

#### Length of side \(CA\):

\[
CA = \sqrt{(-5 - 1)^2 + (3 - (-1))^2}
\]

\[
CA = \sqrt{(-6)^2 + (4)^2}
\]

\[
CA = \sqrt{36 + 16} = \sqrt{52}
\]

### Step 2: Determine the type of triangle

Now, we have the lengths of the sides:

\[
AB = \sqrt{26}, \quad BC = \sqrt{26}, \quad CA = \sqrt{52}
\]

Since \(AB = BC\) and \(AB \neq CA\), the triangle has two equal sides, which means it is an **isosceles triangle**.

### Step 3: Find the length of the median from vertex \(A(1, -1)\)

To find the length of the median joining vertex \(A(1, -1)\) to the midpoint of the opposite side \(BC\), we first need to find the coordinates of the midpoint \(M\) of side \(BC\).

The midpoint \(M\) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

#### Coordinates of midpoint \(M\) of \(BC\):

\[
M = \left( \frac{-5 + 0}{2}, \frac{3 + 4}{2} \right)
\]

\[
M = \left( -\frac{5}{2}, \frac{7}{2} \right)
\]

#### Length of the median \(AM\):

Now, we use the distance formula to find the length of the median \(AM\):

\[
AM = \sqrt{\left( -\frac{5}{2} - 1 \right)^2 + \left( \frac{7}{2} - (-1) \right)^2}
\]

Simplify inside the parentheses:

\[
AM = \sqrt{\left( -\frac{5}{2} - \frac{2}{2} \right)^2 + \left( \frac{7}{2} + \frac{2}{2} \right)^2}
\]

\[
AM = \sqrt{\left( -\frac{7}{2} \right)^2 + \left( \frac{9}{2} \right)^2}
\]

Calculate the squares:

\[
AM = \sqrt{\left( \frac{49}{4} \right) + \left( \frac{81}{4} \right)}
\]

\[
AM = \sqrt{\frac{130}{4}}
\]

\[
AM = \sqrt{32.5}
\]

\[
AM \approx 5.7
\]

### Conclusion

The triangle is **isosceles** (with two equal sides of length \(\sqrt{26}\)). The length of the median from vertex \(A(1, -1)\) to the midpoint of side \(BC\) is approximately \(\boxed{5.7}\).

Would you like more details or have any other questions?

Here are some related questions for further practice:

1. Find the area of the triangle with vertices \((1, -1)\), \((0, 4)\), and \((-5, 3)\).
2. Determine the slope of each side of the triangle.
3. Find the coordinates of the centroid of the triangle.
4. Verify if the triangle is a right triangle.
5. Find the equation of the line passing through points \(A(1, -1)\) and \(B(0, 4)\).

**Tip:** The median of a triangle is a line segment that joins a vertex to the midpoint of the opposite side.

Learn how to identify the type of triangle (scalene, isosceles, or equilateral) with vertices (1, -1), (0, 4), and (-5, 3). Also, find the length of the median from vertex (1, -1) to the midpoint of the opposite side. This problem involves applying the distance formula and the midpoint formula.

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