Question

How many parallelograms are there in the given figure?

• A. $20$
• B. $24$
• C. $28$
• D. $30$

Answer 1

The correct option is D $30$

We shall label the figure as shown below:
The simplest parallelograms are ABML, BCNM, COON, DEFO, OFGH, NOHl, MNIJ and LMJK i.e. $8$ in number.
The parallelograms composed of two components each are ACNL, BDOM, CEFN, LNIK, MOHJ, NFGI, ABJK, BCLJ, CDHI and DEGH i.e. 10 in number.
The parallelograms composed of three components each are ADOL, BEFM, LOHK and MFGJ i.e. $4$ in number.
The parallelograms composed of four components each are AEFL, LFGK, ACIK, BOHJ and CEGI i.e. $5$ in number.
The parallelograms composed of six components each are ADHK and BEGJ i.e. $2$ in number. AEGK is the only parallelograms composed of eight components.
Total number of parallelograms in the figure $=8+10+4+5+2+1=30$
Hence, the answer is (D).