Question

If the vertices of a quadrilateral are $A(-3,1),$ $B(-1,4),$ $C(3,2)$ and $D(t_1,-2)$ and the area of the quadrilateral is $19\text{sq. units}$, then which of the following is/are correct?

• A. $t_1=1$
• B. $t_1=-75$
• C. $t_1=75$
• D. $t_1=-1$

Answer 1

Answer: (A), (B)

The correct options are
A $t_1=1$
B $t_1=-75$

Given vertices of quadrilateral are $A(-3,1),B(-1,4),C(3,2)$ and $D(t_1,-2)$

Now, the area of quadrilateral
$19=\frac{1}{2}\left|\begin{array}{ccccc}{x}_1& x_2& x_3& x_4& x_1\\ y_1& y_2& y_3& y_4& y_1\end{array}\right|\Rightarrow 38=\left|\begin{array}{ccccc}-3& -1& 3& t_1& -3\\ 1& 4& 2& -2& 1\end{array}\right|\Rightarrow 38=|(-12+1)+(-2-12)+(-6-2t_1)+(t_1-6)|\Rightarrow 38=|-25-12-t_1|\Rightarrow t_1+37=\pm 38\Rightarrow t_1=-37\pm 38\therefore t_1=1,-75$