The centroid of a triangle lies at the origin and the coordinates of its two vertices are `(-8, 0)` and (9, 11), the area of the triangle in sq. units is

Options

A. `11//8`
B. `8//11`
C. 88
D. none of these

Answer

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

### Step 1: Identify the Given Information
We know:
- The centroid \( C \) of the triangle is at the origin \( (0, 0) \).
- The coordinates of two vertices are \( P(-8, 0) \) and \( Q(9, 11) \).
- Let the coordinates of the third vertex \( R \) be \( (x_1, y_1) \).

### Step 2: Use the Centroid Formula
The formula for the centroid \( C \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ C = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
Since the centroid \( C \) is at the origin \( (0, 0) \), we can set up the following equations:
\[ \frac{-8 + 9 + x_1}{3} = 0 \quad \text{(1)} \]
\[ \frac{0 + 11 + y_1}{3} = 0 \quad \text{(2)} \]

### Step 3: Solve for \( x_1 \) and \( y_1 \)
From equation (1):
\[ -8 + 9 + x_1 = 0 \implies x_1 = -1 \]

From equation (2):
\[ 0 + 11 + y_1 = 0 \implies y_1 = -11 \]

Thus, the coordinates of the third vertex \( R \) are \( (-1, -11) \).

### Step 4: Use the Area Formula for a Triangle
The area \( A \) of a triangle with vertices at \( (x_1, y_1) \), \( (x_2, y_2) \), 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 coordinates of the vertices \( P(-8, 0) \), \( Q(9, 11) \), and \( R(-1, -11) \):
\[ A = \frac{1}{2} \left| -8(11 - (-11)) + 9(-11 - 0) + (-1)(0 - 11) \right| \]

### Step 5: Simplify the Expression
Calculating each term:
\[ A = \frac{1}{2} \left| -8(11 + 11) + 9(-11) + (-1)(-11) \right| \]
\[ = \frac{1}{2} \left| -8 \times 22 - 99 + 11 \right| \]
\[ = \frac{1}{2} \left| -176 - 99 + 11 \right| \]
\[ = \frac{1}{2} \left| -264 \right| \]
\[ = \frac{1}{2} \times 264 = 132 \]

### Final Answer
The area of the triangle is \( 132 \) square units.

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