Question

An equilateral triangle has each of its sides of length $4cm$. If $(x_i,y_i$ $(i=1,2,3)$ are its vertices then the value of ${\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|}^2$ is

• A. $192$
• B. $768$
• C. $1024$
• D. $128$

Answer 1

The correct option is A $192$

The given determinant $=2($area of triangle$)$

$\therefore {\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|}^2=4($area of $\Delta {)}^2=\left(\frac{\sqrt{3}}{4}{a}^2\right)$

$=\frac{3}{16}\times 4\times 4\times 4\times 4\times 4=48\times 4=192$.