Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Area of a Triangle

The area of a  triangle with vertices   A ( x_1, y_1) ,   B(x_2, Y_2) and   C (x_3,y_3) is denoted by  |\triangle| , where

 \begin{aligned} |\triangle| &=\dfrac{1}{2}\left\{ x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2) \right\}\\&=\dfrac{1}{2} \left\{ x_1(y_2-y_3)-x_2(y_1-y_3)+x_3(y_1-y_2) \right\}\\&=\dfrac{1}{2}\begin{vmatrix}x_1 & x_2 & x_3\\ y_1&y_2 &y_3 \\ 1& 1 & 1\end{vmatrix}\end{aligned}

Example:

Let  \triangle ABC be a triangle with vertices  A(5,3), B(-7,9) and  C (4,5) , then area of  \triangle ABC is 

\begin{aligned}|\triangle|&=\left|\dfrac{1}{2}\{x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\} \right|\\&=\left|\dfrac{1}{2}\{(9-5)+(-7)(5-3)+4(3-9) \} \right|\\&=\left|\dfrac{1}{2} \{ (5\times 4)-7\times 2+4(-6)\}\right|\\&=\left|\dfrac{1}{2}\{20-14-24\} \right|\\&=\left|\dfrac{1}{2}(-18)\right|=|-9|=9\text{ sq. unit}\end{aligned}

Corollary (Collinearity of three points)

The three points  A (x_1,y_1), B (x_2,y_2) and  C (x_3,y_3) will be collinear if the area of the triangle  ABC is zero.

i.e. \begin{vmatrix}x_1 & x_2 & x_3\\ y_1&y_2 &y_3 \\ 1& 1 & 1\end{vmatrix} =0

Example:

The points  A (1,4) , B (2,7) and  C (3,10) are collinear when

\triangle=\begin{vmatrix}x_1 & x_2 & x_3\\ y_1&y_2 &y_3 \\ 1& 1 & 1\end{vmatrix} =0

\implies\begin{vmatrix}1 & 2 & 3\\ 4&7 &10 \\ 1& 1 & 1\end{vmatrix}

 \begin{aligned} &=1(7-10)-2(4-10)+3(4-7)\\&=-3-2(-6)+3(-3)\\&=-3+12-9\\&=0 \end{aligned}

 

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