If the points ( 2a, a), (a, 2a) and (a, a) enclose a triangle of area 72 units, then co-ordinates of the centroid of the triangle may be :

Options

A. `(4, 4)`
B. `(-4,4)`
C. `(12, 12)`
D. `(16, 16)`

Answer

To solve the problem step by step, we need to find the area of the triangle formed by the points (2a, a), (a, 2a), and (a, a) and then determine the coordinates of the centroid.

### Step 1: Identify the vertices of the triangle
The vertices of the triangle are given as:
- \( A(2a, a) \)
- \( B(a, 2a) \)
- \( C(a, a) \)

### Step 2: Use the formula for the area of a triangle
The area \( A \) of a triangle formed by three points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

### Step 3: Substitute the coordinates into the area formula
Substituting the coordinates of points \( A \), \( B \), and \( C \):
\[ \text{Area} = \frac{1}{2} \left| 2a(2a - a) + a(a - a) + a(a - 2a) \right| \]
This simplifies to:
\[ = \frac{1}{2} \left| 2a(a) + a(0) + a(-a) \right| \]
\[ = \frac{1}{2} \left| 2a^2 - a^2 \right| = \frac{1}{2} \left| a^2 \right| = \frac{a^2}{2} \]

### Step 4: Set the area equal to 72
According to the problem, the area is given as 72 units:
\[ \frac{a^2}{2} = 72 \]
Multiplying both sides by 2:
\[ a^2 = 144 \]
Taking the square root:
\[ a = \pm 12 \]

### Step 5: Calculate the coordinates of the centroid
The coordinates of the centroid \( G \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are given by:
\[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]

Substituting the coordinates:
\[ G\left( \frac{2a + a + a}{3}, \frac{a + 2a + a}{3} \right) = G\left( \frac{4a}{3}, \frac{4a}{3} \right) \]

### Step 6: Substitute the values of \( a \)
Using \( a = 12 \):
\[ G\left( \frac{4 \times 12}{3}, \frac{4 \times 12}{3} \right) = G\left( \frac{48}{3}, \frac{48}{3} \right) = G(16, 16) \]

### Final Answer
The coordinates of the centroid of the triangle are \( (16, 16) \).