If `A(6,2),B(4,2)andC(6,4)` are the vertices of `DeltaABC`, then find the length of the median through C.

Answer

To find the length of the median through vertex C in triangle ABC with vertices A(6,2), B(4,2), and C(6,4), we will follow these steps:

### Step 1: Find the midpoint D of side AB
The coordinates of points A and B are given as:
- A(6, 2)
- B(4, 2)

The formula for the midpoint D of a line segment connecting 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 into the formula:
\[ D\left(\frac{6 + 4}{2}, \frac{2 + 2}{2}\right) = D\left(\frac{10}{2}, \frac{4}{2}\right) = D(5, 2) \]

### Step 2: Use the distance formula to find the length of median CD
Now we need to find the length of the median CD, where C is (6, 4) and D is (5, 2).

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

Substituting the coordinates of C and D into the distance formula:
\[ CD = \sqrt{(5 - 6)^2 + (2 - 4)^2} \]

Calculating each part:
\[ CD = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \]

### Final Answer
The length of the median through C is \(\sqrt{5}\).

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