ABCD is a parallelogram. 'O' is an interior point. If ar (AOB) + ar (DOC) = 43 sq units, then find ar (parallelogram ABCD).
Answers
Ar(parallelogram ABCD) = 86 sq units.
Step-by-step explanation:
Given:
ABCD is a parallelogram and O is an interior point.
And. ar (AOB) + ar (DOC) = 43 sq units
Construction: Draw EF and GH through O parallel to AB and AD respectively,
To find:
ar (parallelogram ABCD) = ?
Solution:
In parallelogram FOGA, OA is a diagonal. So,
ar (AOG) = ar (AOF) [1]
In Parallelogram BGOE, OB is a diagonal. So,
ar (AOG) = ar (EOB) [2]
In Parallelogram GHOF, OD is a diagonal. So,
ar (DOH) = ar (DOF) [3]
In Parallelogram HCEO, OC is a diagonal. So,
ar (COH) = ar (COE) [4]
Adding equation (1), (2), (3) and (4), we get,
ar (AOG)+ar (AOG)+ar (DOH)+ar (COH) = ar (AOF)+ar (EOB)+ar (DOF)+ar (COE)
⇒ [ar (AOG)+ar (AOG)] + [ar (DOH)+ar (COH)] = [ ar (AOF)+ar (DOF)] + [ar (EOB)+ar (COE)]
⇒ ar (AOB)+ar (COD) = ar (AOD)+ar (BOC) [shown in the figure] [5]
Now,
Ar(║ABCD) = ar (AOB)+ar (COD) + ar (AOD)+ar (BOC)
= ar (AOB)+ar (COD) + ar (AOB)+ar (COD) [from 5]
= 2 [ar (AOB)+ar (COD)]
= 2 (43) [given]
= 86 sq units.
Ar(parallelogram ABCD) is 86 sq units.
