Question

Find the area of triangle with vertices ( a, b+c) , (b, c+a) and (c, a+b).

 

Answer 1

The area ‘A’ encompassed by three points`(x_1 , y_1) ,(x_1,y_2) "and" (x_3 , y_3) `  is given by the formula,

`A = 1/2 |x_1(y_2 - y_3 ) + x_2 (y_3 -y_1) +x_3 (y_1 - y_2)|`

Here, three points `(x_1 , y_1) ,(x_2,y_2) "and" (x_3 , y_3) `  are \[\left( a, b + c \right), \left( b, c + a \right) and \left( c, a + b \right)\]

Area is as follows: 

\[\left( a, b + c \right), \left( b, c + a \right) and \left( c, a + b \right)\]
\[A = \frac{1}{2}\left| a\left( c + a - a - b \right) + b\left( a + b - b - c \right) + c\left( b + c - c - a \right) \right|\]
\[ = \frac{1}{2}\left| a\left( c - b \right) + b\left( a - c \right) + c\left( b - a \right) \right|\]
\[ = \frac{1}{2}\left| ac - ab + ba - bc + cb - ca \right|\]
\[ = 0\]