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

Answer

To find the length of the median passing through vertex A in triangle ABC with vertices A(-1, 3), B(1, -1), and C(5, 1), we will follow these steps:

### Step 1: Identify the Coordinates of Points
The coordinates of the vertices are:
- A = (-1, 3)
- B = (1, -1)
- C = (5, 1)

### Step 2: Find the Midpoint D of Line Segment BC
The median from vertex A will bisect the line segment BC at point D. We can find the coordinates of D using the midpoint formula:
\[ D\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
where \(B(x_1, y_1) = (1, -1)\) and \(C(x_2, y_2) = (5, 1)\).

Calculating the coordinates of D:
\[ D\left(\frac{1 + 5}{2}, \frac{-1 + 1}{2}\right) = D\left(\frac{6}{2}, \frac{0}{2}\right) = D(3, 0) \]

### Step 3: Use the Distance Formula to Find Length of AD
Now, we will find the length of the median AD using the distance formula:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, we will take:
- \(A(x_1, y_1) = (-1, 3)\)
- \(D(x_2, y_2) = (3, 0)\)

Substituting the coordinates into the distance formula:
\[ AD = \sqrt{(3 - (-1))^2 + (0 - 3)^2} \]
\[ = \sqrt{(3 + 1)^2 + (-3)^2} \]
\[ = \sqrt{4^2 + (-3)^2} \]
\[ = \sqrt{16 + 9} \]
\[ = \sqrt{25} \]
\[ = 5 \]

### Conclusion
The length of the median AD is 5 units.

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