The coordinates of the midpoints of the sides of a triangle `A B C` are `D(2,1),E(5,3)` and `F(3,7)dot` Equation of median of the triangle `A B C` passing through `F` is.

Options

A. `10x+y - 37 = 0`
B. `x+y - 10 = 0`
C. `x - 10y + 67 = 0`
D. None of these

Answer

To find the equation of the median of triangle ABC passing through point F, we will follow these steps:

### Step 1: Identify the coordinates of the midpoints
We are given the coordinates of the midpoints:
- D(2, 1)
- E(5, 3)
- F(3, 7)

### Step 2: Find the coordinates of point O
Point O is the midpoint of line segment DE. We can find the coordinates of O using the midpoint formula:
\[ O\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Substituting the coordinates of D and E:
\[ O\left(\frac{2 + 5}{2}, \frac{1 + 3}{2}\right) = O\left(\frac{7}{2}, 2\right) \]

### Step 3: Use the two points to find the equation of the line
We need to find the equation of the line passing through points F(3, 7) and O\(\left(\frac{7}{2}, 2\right)\). The formula for the equation of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} \]

### Step 4: Substitute the coordinates into the equation
Let:
- \( (x_1, y_1) = (3, 7) \)
- \( (x_2, y_2) = \left(\frac{7}{2}, 2\right) \)

Substituting these values into the line equation:
\[ \frac{y - 7}{2 - 7} = \frac{x - 3}{\frac{7}{2} - 3} \]
This simplifies to:
\[ \frac{y - 7}{-5} = \frac{x - 3}{\frac{1}{2}} \]

### Step 5: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ (y - 7) \cdot \frac{1}{2} = (x - 3)(-5) \]
This simplifies to:
\[ y - 7 = -10(x - 3) \]

### Step 6: Expand and rearrange the equation
Expanding the right side:
\[ y - 7 = -10x + 30 \]
Rearranging gives:
\[ 10x + y - 37 = 0 \]

### Final Result
The equation of the median CF is:
\[ 10x + y - 37 = 0 \]