Question

Consider the point A(a, b + c), B(b, c + a) and (c, a + b) The area of $\Delta ABC$ is

• A. $2(a^2+b^2+c^2)$
• B. $\frac{a^2+b^2+c^2}{6}$
• C. 2(ab + bc + ca)
• D. None of these

Answer 1

Answer: (D)

None of these

Area of a triangle with vertices $(x_1,y_1)$ ; $(x_2,y_2)$ and $(x_3,y_3)$ is $\left|\frac{x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)}{2}\right|$
Hence, substituting the points $(x_1,y_1)=(a,b+c)$ ; $(x_2,y_2)=(b,c+a)$ and $(x_3,y_3)=(c,a+b)$ in the area formula, we get $\left|\frac{a(c+a-a-b)+b(a+b-b-c)+c(b+c-c-a)}{2}\right|=\left|\frac{a(c-b)+b(a-c)+c(b-a)}{2}\right|=\left|\frac{ac-ab+ab-bc+bc-ac}{2}\right|=0$