A triangle has coordinates A(1,1),B(4,2),C(3,5). What is the area of triangle ABC?

Guest Sep 5, 2020

#1**+2 **

A triangle has coordinates A(1,1),B(4,2),C(3,5). What is the area of triangle ABC?

**Hello Guest!**

\(y=m(x-x_1)+y_1\)

\(m_a=\frac{5-2}{3-4}=-3\\ m_b=\frac{5-1}{3-1}=2\\ m_c=\frac{2-1}{4-1}=\frac{1}{3}\)

\(y_a=-3(x-4)+2=-3x+14\\ y_b=2(x-1)+1=2x-1\\ y_c=\frac{1}{3}(x-1)+1=\frac{1}{3}x+\frac{2}{3}\)

\(A_{ABC}=\int _3^4(-3x+14)+\int_1^3(2x-1)-\int_1^4(\frac{1}{3}x+\frac{2}{3})\)

**\(A_{ABC}=|-\frac{3}{2}x^2+14x|_3^4+|x^2-x|_1^3-|\frac{1}{6}x^2+\frac{2}{3}x|_1^4\)**

**\(A_{ABC}=(-24+56+13.5-42)+(9-3-1+1) \)**

** \(-(\frac{16}{6}+\frac{8}{3}-\frac{1}{6}-\frac{2}{3})=5\)**

**\(\large A_{ABC}=5\)**

** !**

asinus Sep 5, 2020