Question

$A(p,0),B(4,0),C(5,6)$ and $D(1,4)$ are the vertices of a quadrilateral $ABCD$.

If two sides of the quadrilateral are equal; area of the quadrilateral is

• A. $\frac{25}{3}$
• B. $11$
• C. $\frac{58}{3}$
• D. none of these

Answer 1

The correct option is C $\frac{58}{3}$

$(AB{)}^2=(AD{)}^2$
$\Rightarrow (p-4{)}^2=(p-1{)}^2+4^2$
$\Rightarrow p=\frac{-1}{6}$

Area of the quadrilateral ABCD $=$ Area of $\Delta ABD$+Area of $\Delta BDC$
$=\left|\begin{array}{c}\frac{1}{2}\left|\begin{array}{ccc}\frac{-1}{6}& 0& 1\\ 4& 0& 1\\ 1& 4& 1\end{array}\right|\end{array}\right|+\left|\begin{array}{c}\frac{1}{2}\left|\begin{array}{ccc}4& 0& 1\\ 1& 4& 1\\ 5& 6& 1\end{array}\right|\end{array}\right|$

$=\frac{25}{3}+11=\frac{58}{3}$