If `((3)/(2),0), ((3)/(2), 6)` and `(-1, 6)` are mid-points of the sides of a triangle, then find
Centroid of the triangle

Answer

To find the centroid of the triangle given the midpoints of its sides, we will follow these steps:

### Step 1: Identify the midpoints and set up equations
We are given the midpoints of the triangle:
- D = \((\frac{3}{2}, 0)\)
- E = \((\frac{3}{2}, 6)\)
- F = \((-1, 6)\)

Let the vertices of the triangle be A, B, and C with coordinates:
- A = \((x_1, y_1)\)
- B = \((x_2, y_2)\)
- C = \((x_3, y_3)\)

Using the midpoint formula, we can set up the following equations based on the midpoints:

1. For midpoint D (between A and B):
\[ \frac{x_1 + x_2}{2} = \frac{3}{2} \implies x_1 + x_2 = 3 \quad \text{(Equation 1)} \]
\[ \frac{y_1 + y_2}{2} = 0 \implies y_1 + y_2 = 0 \quad \text{(Equation 2)} \]

2. For midpoint E (between B and C):
\[ \frac{x_2 + x_3}{2} = \frac{3}{2} \implies x_2 + x_3 = 3 \quad \text{(Equation 3)} \]
\[ \frac{y_2 + y_3}{2} = 6 \implies y_2 + y_3 = 12 \quad \text{(Equation 4)} \]

3. For midpoint F (between A and C):
\[ \frac{x_1 + x_3}{2} = -1 \implies x_1 + x_3 = -2 \quad \text{(Equation 5)} \]
\[ \frac{y_1 + y_3}{2} = 6 \implies y_1 + y_3 = 12 \quad \text{(Equation 6)} \]

### Step 2: Solve the equations for \(x_1, x_2, x_3\)
Now, we will solve the equations for \(x_1, x_2, x_3\).

From Equations 1, 3, and 5:
1. \(x_1 + x_2 = 3\)
2. \(x_2 + x_3 = 3\)
3. \(x_1 + x_3 = -2\)

Adding these three equations:
\[ (x_1 + x_2) + (x_2 + x_3) + (x_1 + x_3) = 3 + 3 - 2 \]
This simplifies to:
\[ 2(x_1 + x_2 + x_3) = 4 \implies x_1 + x_2 + x_3 = 2 \]

### Step 3: Solve the equations for \(y_1, y_2, y_3\)
Now, we will solve the equations for \(y_1, y_2, y_3\).

From Equations 2, 4, and 6:
1. \(y_1 + y_2 = 0\)
2. \(y_2 + y_3 = 12\)
3. \(y_1 + y_3 = 12\)

Adding these three equations:
\[ (y_1 + y_2) + (y_2 + y_3) + (y_1 + y_3) = 0 + 12 + 12 \]
This simplifies to:
\[ 2(y_1 + y_2 + y_3) = 24 \implies y_1 + y_2 + y_3 = 12 \]

### Step 4: Find the centroid
The centroid \(G\) of the triangle is given by the formula:
\[ G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]

Substituting the values we found:
\[ G = \left(\frac{2}{3}, \frac{12}{3}\right) = \left(\frac{2}{3}, 4\right) \]

### Final Answer
The centroid of the triangle is \(\left(\frac{2}{3}, 4\right)\).

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