Question

$ABCD$ is a quadrilateral and let $\overline{AB}=\overline{a},\overline{AD}=\overline{b},\overline{AC}=m\overline{a}+n\overline{b}$. Then the area of the quadrilateral is

• A. $(m+n)(\overline{a}\times \overline{b})$
• B. $2|m+n|(\overline{a}\times \overline{b})$
• C. $\frac{1}{2}|m+n||\overline{a}\times \overline{b}|$
• D. $3(m+n)|\overline{a}\times \overline{b}|$

Answer 1

The correct option is C $\frac{1}{2}|m+n||\overline{a}\times \overline{b}|$

$\overrightarrow{AD}=\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}$
$\overrightarrow{b}-\overrightarrow{a}=\overrightarrow{BD}$
$\overrightarrow{n}=\overrightarrow{BD}\times \overrightarrow{AC}$
$=(\overrightarrow{b}-\overrightarrow{a})\times (m\overrightarrow{a}+n\overrightarrow{b})$
$=m\overrightarrow{b}\times \overrightarrow{a}-n\overrightarrow{a}\times \overrightarrow{b}$
$|\overrightarrow{n}|=(m+n)(\overrightarrow{a}\times \overrightarrow{b})$
Area of quadrilateral $=\frac{1}{2}(m+n)|\overrightarrow{a}\times \overrightarrow{b}|$