The centroid of a triangle is (1,4) and the co-ordinate of its two vertices are (4,-3) and (-9,7). Then the area of the triangle is

Options

A. `183//2`
B. `-183//2`
C. `183`
D. None of these

Answer

To find the area of the triangle given the centroid and two vertices, we can follow these steps:

### Step 1: Understand the Centroid Formula
The centroid \( G \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) is given by:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]
Given that the centroid \( G \) is \( (1, 4) \) and the coordinates of two vertices are \( B(4, -3) \) and \( C(-9, 7) \), we can denote the third vertex \( A(x, y) \).

### Step 2: Set Up the Equations
From the centroid formula, we can set up the following equations:
1. For the x-coordinate:
\[ \frac{x + 4 - 9}{3} = 1 \]
2. For the y-coordinate:
\[ \frac{y - 3 + 7}{3} = 4 \]

### Step 3: Solve for \( x \)
From the first equation:
\[ \frac{x - 5}{3} = 1 \]
Multiplying both sides by 3:
\[ x - 5 = 3 \]
Adding 5 to both sides:
\[ x = 8 \]

### Step 4: Solve for \( y \)
From the second equation:
\[ \frac{y + 4}{3} = 4 \]
Multiplying both sides by 3:
\[ y + 4 = 12 \]
Subtracting 4 from both sides:
\[ y = 8 \]

### Step 5: Identify the Vertices
Now we have the coordinates of the vertices:
- \( A(8, 8) \)
- \( B(4, -3) \)
- \( C(-9, 7) \)

### Step 6: Calculate the Area of the Triangle
The area \( A \) of triangle \( ABC \) 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 coordinates:
\[ A = \frac{1}{2} \left| 8(-3 - 7) + 4(7 - 8) + (-9)(8 + 3) \right| \]
Calculating each term:
1. \( 8(-10) = -80 \)
2. \( 4(-1) = -4 \)
3. \( -9(11) = -99 \)

Now substituting back:
\[ A = \frac{1}{2} \left| -80 - 4 - 99 \right| = \frac{1}{2} \left| -183 \right| = \frac{183}{2} \]

### Final Answer
The area of the triangle is:
\[ \frac{183}{2} \text{ square units} \]

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