Vertices of a `DeltaABC` are `A(2,2),B(-4,-4),C(5,-8)`. Then length of the median through C is

Options

A. `sqrt(65)`
B. `sqrt(117)`
C. `sqrt(85)`
D. `sqrt(113)`

Answer

To find the length of the median through vertex C of triangle ABC with vertices A(2, 2), B(-4, -4), and C(5, -8), we can follow these steps:

### Step 1: Find the midpoint D of segment AB
The coordinates of points A and B are given as:
- A(2, 2)
- B(-4, -4)

The formula for the midpoint D of a line segment connecting two points (x1, y1) and (x2, y2) is:
\[ D\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]

Substituting the coordinates of A and B:
\[ D\left(\frac{2 + (-4)}{2}, \frac{2 + (-4)}{2}\right) = D\left(\frac{-2}{2}, \frac{-2}{2}\right) = D(-1, -1) \]

### Step 2: Use the distance formula to find the length of median CD
The coordinates of point C are given as:
- C(5, -8)

The distance formula between two points (x1, y1) and (x2, y2) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Now, we will use this formula to find the length of median CD:
\[ CD = \sqrt{(5 - (-1))^2 + (-8 - (-1))^2} \]
This simplifies to:
\[ CD = \sqrt{(5 + 1)^2 + (-8 + 1)^2} = \sqrt{(6)^2 + (-7)^2} \]
Calculating the squares:
\[ CD = \sqrt{36 + 49} = \sqrt{85} \]

### Conclusion
The length of the median through C is:
\[ \sqrt{85} \]