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

Options

A. `sqrt65`
B. `sqrt117`
C. `sqrt85`
D. `sqrt113`

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 will follow these steps:

### Step 1: Find the Midpoint of AB
The median from vertex C to side AB will intersect AB at its midpoint. The formula for the midpoint M(x, y) of a line segment connecting two points A(x1, y1) and B(x2, y2) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Here, A(2, 2) and B(-4, -4).

Calculating the midpoint:
\[ M = \left( \frac{2 + (-4)}{2}, \frac{2 + (-4)}{2} \right) = \left( \frac{-2}{2}, \frac{-2}{2} \right) = (-1, -1) \]

### Step 2: Calculate the Length of the Median
Now, we need to find the length of the median from point C(5, -8) to the midpoint M(-1, -1). The distance formula between two points C(x1, y1) and M(x2, y2) is:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates of C and M:
\[ \text{Distance} = \sqrt{((-1) - 5)^2 + ((-1) - (-8))^2} \]
\[ = \sqrt{(-6)^2 + (7)^2} = \sqrt{36 + 49} = \sqrt{85} \]

### Step 3: Conclusion
Thus, the length of the median through C is:
\[ \sqrt{85} \]