If $P(2,\ 1),\ Q(4,\ 2),\ R(5,\ 4)$ and $S(3,\ 3)$ are vertices of a quadrilateral $PQRS$, find the area of the quadrilateral $PQRS$.


Given: $P(2,\ 1),\ Q(4,\ 2),\ R(5,\ 4)$ and $S(3,\ 3)$ are vertices of a quadrilateral $PQRS$.

To do: To find the area of the quadrilateral $PQRS$. 

Solution:

The vertices of the quadrilateral PQRS are $P( 2,\ 1),\ Q( 4,\ 2),\ R( 5,\ 4),\ S( 3,\ 3)$

Area of quadrilateral PQRS$=Area( \vartriangle PQR)+Area( \vartriangle PSR)$

Area of $\vartriangle PQR=\frac{1}{2}[x_1( y_2-y_3)+x_2( y_3-y_1)+x_3( y_1-y_2)]$

$=\frac{1}{2}[2( 2-4)+4( 4-1)+5( 1-2)]$

$=\frac{1}{2}[2( -2)+4( 3)+5(-1)]$

$=\frac{1}{2}[-4+12-5]$

$=\frac{1}{2}[12-9]$

$=\frac{1}{2}[3]$

$=1.5$

$\Rightarrow$ Area of triangle $PQR=1.5\ sq.\ units$

Also, Area of triangle PSR $=\frac{1}{2}[2( 3-4)+3( 4-1)+5( 1-3)]$

$=\frac{1}{2}[2( -1)+3( 3)+5( -2)]$

$=\frac{1}{2}[-2+9-10]$

$=\frac{1}{2}[-3]$

Area of triangle PSR $=-1.5\ sq.\ units$    

Therefore,    

Area of quadrilateral PQRS$= 1.5 + 1.5=3\ sq.\ units$

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Updated on: 10-Oct-2022

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