question_answer

If \[P(-5,-3),Q(-4,-6),R(2,-3)\] and \[S(1,2)\] are the vertices of a quadrilateral PQRS, find its area.

Answer:

We have \[P(-5,-3),Q(-4,-6),R(2,-3)\]and \[S(1,2)\] are the vertices of a quadrilateral PQRS.

Join P and R. Then,

Area of quad PQRS = (Area of \[\Delta \text{ }PQR\]) + (Area of\[\Delta \,PRS\])

Area of \[\Delta \text{ }PQR\]

\[=\frac{1}{2}|-5(-6+3)-4(-3+3)+2(-3+6)|\]

\[=\frac{1}{2}|-5(-3)-4(0)+2(3)|\]

=\[\frac{1}{2}|15+6|=\frac{21}{2}\] sq. units

And, area of \[\Delta \text{ }PRS\]

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

\[=\frac{1}{2}=|-5(-5)+2(5)+1(0)|\]

\[=\frac{1}{2}|25+10|=\frac{35}{2}\] sq. units

Hence, area of quad. \[PQRS=\text{Area of }\Delta \text{ }PQR+\text{Area of }\Delta \text{ }PRS\]

\[=\left( \frac{21}{2}+\frac{35}{2} \right)\] sq. units

\[=\frac{56}{2}\] sq. units

\[=28\] sq. units