If the points (2k,k),(k,2k) and (k,k) with `kgt0` enclose a triangle of area 18 square units then the centroid of triangle is equal to

Options

A. (8,8)
B. (4,4)
C. (-4,-4)
D. `(4sqrt(2),4sqrt(2))`

Answer

To solve the problem, we need to find the centroid of the triangle formed by the points \( (2k, k) \), \( (k, 2k) \), and \( (k, k) \) given that the area of the triangle is 18 square units.

### Step 1: Calculate the area of the triangle using the determinant formula

The area \( A \) of a triangle formed by the points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be calculated using the formula:

\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Substituting the points \( (2k, k) \), \( (k, 2k) \), and \( (k, k) \):

\[ A = \frac{1}{2} \left| 2k(2k - k) + k(k - k) + k(k - 2k) \right| \]

### Step 2: Simplify the area expression

Calculating the terms inside the absolute value:

\[ A = \frac{1}{2} \left| 2k(k) + k(0) + k(-k) \right| \]
\[ = \frac{1}{2} \left| 2k^2 - k^2 \right| \]
\[ = \frac{1}{2} \left| k^2 \right| \]

Since \( k > 0 \), we can drop the absolute value:

\[ A = \frac{1}{2} k^2 \]

### Step 3: Set the area equal to 18 and solve for \( k \)

Given that the area is 18 square units:

\[ \frac{1}{2} k^2 = 18 \]

Multiplying both sides by 2:

\[ k^2 = 36 \]

Taking the square root:

\[ k = 6 \quad (\text{since } k > 0) \]

### Step 4: Find the coordinates of the vertices

Now we substitute \( k = 6 \) back into the points:

1. \( (2k, k) = (12, 6) \)
2. \( (k, 2k) = (6, 12) \)
3. \( (k, k) = (6, 6) \)

### Step 5: Calculate the centroid of the triangle

The centroid \( (G_x, G_y) \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:

\[ G_x = \frac{x_1 + x_2 + x_3}{3}, \quad G_y = \frac{y_1 + y_2 + y_3}{3} \]

Substituting the vertices:

\[ G_x = \frac{12 + 6 + 6}{3} = \frac{24}{3} = 8 \]
\[ G_y = \frac{6 + 12 + 6}{3} = \frac{24}{3} = 8 \]

Thus, the centroid of the triangle is \( (8, 8) \).

### Final Answer

The centroid of the triangle is \( (8, 8) \).

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