If a triangle `A B C ,A-=(1,10),` circumcenter `-=(-1/3,2/3),` and orthocentre `-=((11)/3,4/3)` , then the coordinates of the midpoint of the side opposite to `A` are

Options

A. `(1,-11//3)`
B. `(1//5)`
C. `(1,-3)`
D. `(1,6)`

Explanation

To find the coordinates of the midpoint of the side opposite to vertex A in triangle ABC, we can follow these steps:

### Step 1: Identify Given Points
We are given:
- Vertex A = (1, 10)
- Circumcenter O = (-1/3, 2/3)
- Orthocenter H = (11/3, 4/3)

### Step 2: Find the Centroid G
The centroid G divides the line segment joining the circumcenter O and the orthocenter H in the ratio 2:1. We can use the section formula to find the coordinates of G.

Using the section formula:
\[ G = \left( \frac{m \cdot x_2 + n \cdot x_1}{m+n}, \frac{m \cdot y_2 + n \cdot y_1}{m+n} \right) \]
where \( m = 1 \) and \( n = 2 \) (since G divides OH in the ratio 2:1).

Substituting the coordinates:
- \( O = (-1/3, 2/3) \)
- \( H = (11/3, 4/3) \)

Calculating the x-coordinate of G:
\[ G_x = \frac{1 \cdot \frac{11}{3} + 2 \cdot \left(-\frac{1}{3}\right)}{1 + 2} = \frac{\frac{11}{3} - \frac{2}{3}}{3} = \frac{\frac{9}{3}}{3} = 1 \]

Calculating the y-coordinate of G:
\[ G_y = \frac{1 \cdot \frac{4}{3} + 2 \cdot \frac{2}{3}}{1 + 2} = \frac{\frac{4}{3} + \frac{4}{3}}{3} = \frac{\frac{8}{3}}{3} = \frac{8}{9} \]

Thus, the coordinates of the centroid G are:
\[ G = \left(1, \frac{8}{9}\right) \]

### Step 3: Use the Centroid to Find Midpoint D
Let D be the midpoint of side BC. The centroid G divides the median AD in the ratio 2:1.

Using the formula for the centroid again:
\[ G = \left( \frac{x_1 + x_2}{3}, \frac{y_1 + y_2}{3} \right) \]
where \( A = (1, 10) \) and \( D = (x, y) \).

Setting up the equations:
1. For x-coordinates:
\[ \frac{x + 1}{3} = 1 \implies x + 1 = 3 \implies x = 2 \]

2. For y-coordinates:
\[ \frac{y + 10}{3} = \frac{8}{9} \implies y + 10 = \frac{8}{3} \implies y = \frac{8}{3} - 10 = \frac{8}{3} - \frac{30}{3} = -\frac{22}{3} \]

Thus, the coordinates of D (the midpoint of BC) are:
\[ D = \left(2, -\frac{22}{3}\right) \]

### Final Answer
The coordinates of the midpoint of the side opposite to A are:
\[ (2, -\frac{22}{3}) \]