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Q.

State true or false.


In the given figure, O is any point inside a parallelogram ABCD. Then,


ar ΔOAB  + ar ΔOCD  =   1 2  ar(Parallelogram ABCD).  


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a

True

b

False 

answer is A.

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Detailed Solution

It is given that O is any point inside a parallelogram ABCD.
Question ImageWe have to check if ar(∆OAB) + ar(∆OCD) = 1 2   ar(parallelogram ABCD).
Question ImageWhen a triangle and parallelogram are aligned on the same base and between the same parallel lines, then the area of the triangle is equal to half of the area of the parallelogram.
Considering a parallelogram and constructing PQ||AB, RS||AD as depicted in the figure below.
Considering ∆OAB and parallelogram ABQP shown in the figure. They have the same base AB and lie between parallel lines AB and PQ.
Area of ∆OAB = 1 2    area of parallelogram ABQP…...equation (i)
Area of ∆OCD = 1 2   area of parallelogram PQCD…...equation (ii)
 Observe and add equations (i) and (ii).
⇒   Area of ∆OAB+ area of ∆OCD= 1 2    area of parallelogram ABQP+ 1 2   area of parallelogram PQCD.
⇒   Area of ∆OAB+ area of ∆OCD= 1 2  area of parallelogram ABCD  Hence proved, Area of ∆OAB+ area of ∆OCD= 1 2  area of parallelogram ABCD.
Thus, the statement is true.
Therefore, option 1 is correct.
 
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