If centroid of a triangle be (1, 4) and the coordinates of its any two vertices are (4, -8) and (-9, 7), find the area of the triangle.

Answer

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

### Step 1: Identify the given information
We have:
- Centroid \( G(1, 4) \)
- Vertex \( A(4, -8) \)
- Vertex \( B(-9, 7) \)
- Let the third vertex be \( C(h, k) \).

### Step 2: Use the centroid formula
The formula for 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:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]

Substituting the coordinates of the known vertices and the centroid:
\[ 1 = \frac{4 + (-9) + h}{3} \]
\[ 4 = \frac{-8 + 7 + k}{3} \]

### Step 3: Solve for \( h \)
From the first equation:
\[ 1 = \frac{4 - 9 + h}{3} \]
Multiplying both sides by 3:
\[ 3 = 4 - 9 + h \]
\[ h = 3 + 5 = 8 \]

### Step 4: Solve for \( k \)
From the second equation:
\[ 4 = \frac{-8 + 7 + k}{3} \]
Multiplying both sides by 3:
\[ 12 = -8 + 7 + k \]
\[ k = 12 + 1 = 13 \]

### Step 5: Identify the coordinates of the third vertex
Now we have the coordinates of the third vertex \( C(8, 13) \).

### Step 6: Use the area formula for a triangle
The area \( A \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Substituting the coordinates \( A(4, -8) \), \( B(-9, 7) \), and \( C(8, 13) \):
\[ A = \frac{1}{2} \left| 4(7 - 13) + (-9)(13 + 8) + 8(-8 - 7) \right| \]

### Step 7: Calculate the area
Calculating each term:
- \( 4(7 - 13) = 4 \times (-6) = -24 \)
- \( -9(13 + 8) = -9 \times 21 = -189 \)
- \( 8(-8 - 7) = 8 \times (-15) = -120 \)

Putting it all together:
\[ A = \frac{1}{2} \left| -24 - 189 - 120 \right| \]
\[ = \frac{1}{2} \left| -333 \right| = \frac{333}{2} \]

### Final Answer
The area of the triangle is \( \frac{333}{2} \) square units.

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