Question

Calculate the area of a quadrilateral PQRS whose diagonal $PR = 20cm$and the lengths of perpendicular from Q and S on PR is $12cm\,\,and\,\,10cm$respectively.

Answer 1

Hint:We will use the properties of triangles in a quadrilateral and areas of triangles to find the area of the quadrilateral ABCD.

Complete answer:
Here, it is given that PQRS is a quadrilateral in which diagonal $PR = 20cm$and length of perpendiculars from point Q and S are $12cm\,\,and\,\,10cm$respectively.

From the figure it is very much clear that diagonal PR divides quadrilateral PQRS in two triangles.
Therefore, its area can be calculated by calculating the area of two triangles PQR and PSR and then adding them.
Area of triangle PQR is given as: $\dfrac{1}{2} \times base \times altitude$
Substituting, value of base and altitude in above. We have,
$
  \Rightarrow ar\left( {\Delta PQR} \right) = \dfrac{1}{2} \times 20 \times 12 \\
   \Rightarrow ar\left( {\Delta PQR} \right) = 10 \times 12 \\
  \Rightarrow ar\left( {\Delta PQR} \right) = 120 \\
 $
Therefore, area of a triangle PQR is $120\,\,c{m^2}$………………..(i)
Area of triangle PSR is given as: $\dfrac{1}{2} \times base \times altitude$
Substituting, value of base and altitude in above. We have,
$
  \Rightarrow ar\left( {\Delta PSR} \right) = \dfrac{1}{2} \times 20 \times 10 \\
   \Rightarrow ar\left( {\Delta PSR} \right) = 10 \times 10 \\
  \Rightarrow ar\left( {\Delta PSR} \right) = 100 \\
 $
Therefore, area of a triangle PSR is $100\,\,c{m^2}$………………..(ii)
Adding (i) and (ii) we have
$
  \Rightarrow ar\left( {\Delta PSR} \right) + ar\left( {\Delta PQR} \right) = 100 + 120 \\
  \Rightarrow ar\left( {Qd.\,PQRS} \right) = 220 \\
 $
Therefore, from above we see that area of a quadrilateral PQRS is $220\,c{m^2}$

Note: We can also find the area of a quadrilateral in another way. In this way instead of dividing a quadrilateral in two parts. We directly apply a mensuration formula to find the area of the quadrilateral. Which is$\dfrac{1}{2}\left( {{p_1} + {p_2}} \right) \times D$, where ${p_1}\,and\,\,{p_2}$are perpendiculars and ‘D’ is the longest diagonal of the quadrilateral.