Question

show that the two diagonals of a parallelogram divide it in to four triangle of equal area. ​

Answer 1

Given :

• A parallelogram ABCD with diagonals AC and AD .

To prove :

• Diagonals divide parallelogram into four triangles of equal area .

Proof :

Since , diagonals of parallelogram bisect each other ,

• AO = OC
• BO = OD

• •°• , O is the mid point of both AC and BD .

Hence , BO is median of ∆ ABC , DO is median of ∆ADC , AO is median of ∆ABD and CO of ∆BCD.

Now , we know that median divides triangle into two equal areas :

•°• ar ( ∆AOB) = ar (∆ BOC) [in ∆ ABC] ____(1)

ar (∆ AOD) = ar (∆COD) [in ∆ ADC] _____(2)

ar (∆AOB) = ar (∆ AOD) [in ∆ ABD] ____(3)

Combining equation (1) , (2) and (3)

• ar (∆ AOB ) = ar ( ∆BOC) = ar (∆COD) = ar ( ∆AOD)

Hence , diagonals divide parallelogram into four triangles of equal areas.