Let ABC be an acute- angled triangle and AD, BE, and CF be its medians, where E and F are at (3,4) and (1,2) respectively. The centroid of `DeltaABC` `G(3,2)`.
The coordinates of point D is ____________

Explanation

To find the coordinates of point D in triangle ABC where AD, BE, and CF are medians, we can follow these steps:

### Step 1: Understand the properties of the centroid
The centroid (G) of a triangle is the average of the coordinates of its vertices. For triangle ABC, the centroid is given as G(3, 2).

### Step 2: Write the formula for the centroid
The coordinates of the centroid G of triangle DEF can be calculated using the formula:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of points D, E, and F respectively.

### Step 3: Substitute the known values
We know the coordinates of points E and F:
- E(3, 4)
- F(1, 2)

Let the coordinates of point D be (α, β). Therefore, we can write:
\[ G = \left( \frac{\alpha + 3 + 1}{3}, \frac{\beta + 4 + 2}{3} \right) \]

### Step 4: Set up equations based on the centroid
Since the centroid G is given as (3, 2), we can set up the following equations:
1. \(\frac{\alpha + 4}{3} = 3\)
2. \(\frac{\beta + 6}{3} = 2\)

### Step 5: Solve for α
From the first equation:
\[ \frac{\alpha + 4}{3} = 3 \]
Multiplying both sides by 3:
\[ \alpha + 4 = 9 \]
Subtracting 4 from both sides:
\[ \alpha = 5 \]

### Step 6: Solve for β
From the second equation:
\[ \frac{\beta + 6}{3} = 2 \]
Multiplying both sides by 3:
\[ \beta + 6 = 6 \]
Subtracting 6 from both sides:
\[ \beta = 0 \]

### Step 7: Write the coordinates of point D
Thus, the coordinates of point D are:
\[ D(5, 0) \]

### Final Answer
The coordinates of point D are (5, 0).

---