Area of triangle formed by the points `A(a, b+c)` , `B(b, c+a)` and `C(c, a+b)` depends upon

Options

A. a
B. b
C. c
D. independent from a,b,and c

Answer

To find the area of the triangle formed by the points \( A(a, b+c) \), \( B(b, c+a) \), and \( C(c, a+b) \), we can use the formula for the area of a triangle given by three vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\):

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

### Step-by-Step Solution:

1. **Identify the coordinates:**
- \( A(a, b+c) \) corresponds to \( (x_1, y_1) = (a, b+c) \)
- \( B(b, c+a) \) corresponds to \( (x_2, y_2) = (b, c+a) \)
- \( C(c, a+b) \) corresponds to \( (x_3, y_3) = (c, a+b) \)

2. **Substitute the coordinates into the area formula:**
\[ \text{Area} = \frac{1}{2} \left| a((c+a) - (a+b)) + b((a+b) - (b+c)) + c((b+c) - (c+a)) \right| \]

3. **Simplify each term:**
- For the first term:
\[ a((c+a) - (a+b)) = a(c - b) \]
- For the second term:
\[ b((a+b) - (b+c)) = b(a - c) \]
- For the third term:
\[ c((b+c) - (c+a)) = c(b - a) \]

4. **Combine the terms:**
\[ \text{Area} = \frac{1}{2} \left| a(c - b) + b(a - c) + c(b - a) \right| \]

5. **Expand and rearrange:**
\[ = \frac{1}{2} \left| ac - ab + ab - bc + bc - ac \right| \]
\[ = \frac{1}{2} \left| 0 \right| = 0 \]

6. **Conclusion:**
The area of the triangle formed by the points \( A \), \( B \), and \( C \) is \( 0 \). This indicates that the points are collinear, and thus the area does not depend on the values of \( a \), \( b \), and \( c \).