If A(1,-1,2),B(5,7,-6),C(3,4,-10) and D(-1,-4,-2) are the vertices of a quadrilateral ABCD…

Question

If $A(1,-1,2),B(5,7,-6),C(3,4,-10)$ and $D(-1,-4,-2)$ are the vertices of a quadrilateral ABCD, then its area is :

Accepted Answer

Area of quadrilateral $ABCD=$ area of $	riangleABC+$ area of $	riangleADC$

In $	riangleABC$

$\begin{gathered} \overset{	o \leftarrow }{AB}=4,8,-8 \ \overset{	o \leftarrow }{BC}=-2,-3,-4 \ \overset{	o \leftarrow }{AB}	imes \overset{	o \leftarrow }{BC}=|\begin{array}{ccc}\hat{i} \quad  \hat{j} \quad  \hat{k} \ 4 \quad  8 \quad  -8 \ -2 \quad  -3 \quad  -4\end{array}| \ =-56\hat{i}+32\hat{j}+4\hat{k} \end{gathered}$

Area of $	riangleABC=\frac{1}{2}|\overset{	o \leftarrow }{AB}	imes \overset{	o \leftarrow }{BC}|$

$\begin{gathered} =\frac{1}{2}\sqrt{{56}^{2}+{32}^{2}+{4}^{2}} \ =\frac{1}{2}\sqrt{4176}=\frac{12\sqrt{29}}{2}=6\sqrt{29} \end{gathered}$

In $	riangleADC=\overset{	o \leftarrow }{AD}=-2,-3,-4$

$\overset{	o \leftarrow }{DC}=4,8,-8$

$\overset{	o \leftarrow }{AD}	imes \overset{	o \leftarrow }{DC}=\begin{vmatrix} \hat{i} \quad  \hat{j} \quad  \hat{k} \ -2 \quad  -3 \quad  -4 \ 4 \quad  8 \quad  -8 \end{vmatrix}$

$=56\hat{i}-32\hat{j}-4k$

$\frac{1}{2}|\overset{	o \leftarrow }{AD}	imes \overset{	o \leftarrow }{DC}|$

$\begin{gathered} =\frac{1}{2}\sqrt{4176} \ =6\sqrt{29} \end{gathered}$

Area of $ABCD=6\sqrt{29}+6\sqrt{29}$

$=12\sqrt{29}$

Solutions · 1