The figure may be labelled as shown.

The simplest squares are EFRQ, MQYX, QRZY, RNSZ, LXWK, XYA₁W, YZB₁A₁, ZSTB₁, SGHT, WA₁VP, A₁B₁UV, B₁TOU and VUIJ i.e. 13 in number.

The squares having two components each are AEYL, FBGZ, KA₁JD and B₁HCI i.e. 4 in number.

The squares having four components each are MRB₁W, QNTA₁ XZUP and YSOV i.e. 4 in number.

The squares having seven components each are AFB₁K, EBHA₁ LZID and YGCJ i.e. 4 in number.

There is only one square i.e. MNOP composed of nine components.

There is only one square i.e. ABCD composed of seventeen components.

There are 13+ 4 + 4 + 4+1 + 1 = 27 squares in the figure.

The figure may be labelled as shown.

The horizontal lines are AE and JF i.e. 2 in number. The vertical lines are AJ, CH and EF i.e. 3 in number.

The slanting lines are AG, BF, JD, IE, AB, DE, JI and FG i.e. 8 in number.

Total number of straight lines needed to construct the figure = 2 + 3 + 8 = 13.

The figure may be labelled as shown.

The simplest rectangles are ABJI, BCKJ, IJFG and JKEF i.e. 4 in number.

The rectangles composed of two components each are ACKI, BCEF, IKEG and ABFG i.e. 4 in number.

The only rectangle composed of four components is ACEG.

Thus, there are 4 + 4 + 1 = 9 rectangles in the given figure.

The figure may be labelled as shown.

Rectangles:

The simplest rectangles are CVSR, VETS, RSWM and STKW i.e. 4 in number.

The rectangles composed of two components each are CETR, VEKW, RTKM and CVWM i.e. 4 in number.

The rectangles composed of three components each are ACRP, PRMO, EGHT and THIK i.e. 4 in number.

The rectangles composed of four components each are CEKM, AVSP,PSWO, VGHS and SHIW i.e. 5 in number.

The rectangles composed of five components each are AETP, PTKO, CGHR and RHIM i.e. 4 in number.

The rectangles composed of six components each are ACMO and EGIK i.e. 2 in number.

The rectangles composed of eight components each are AGHP, PHIO, AVWO and VGIW i.e. 4 in number.

The rectangles composed of ten components each are AEKO and CGIM i.e. 2 in number.

AGIO is the only rectangle having sixteen components.

Total number of rectangles in the given figure

= 4 + 4 + 4 + 5 + 4 + 2 + 4 + 2 + 1 = 30.

Hexagons:

The hexagons in the given figure are CDEKLM, CEUKMQ, CFHJMQ, BEUKNP and BFHJNP.

So, there are 5 hexagons in the given figure.

The figure may be labelled as shown.

Triangles :

The simplest triangles are AEI, EOI, OHI, HAI, EBJ, BFJ, FOJ, OEJ, HOL, OGL, GDL, DHL, OFK, FCK, CGK and GOK i.e. 16 in number.

The triangles composed of two components each are HAE, AEO, EOH, OHA, OEB, EBF, BFO, FOE, DHO, HOG, OGD, GDH, GOF, OFC, FCG and CGO i.e. 16 in number.

The triangles composed of four components each are HEF, EFG, FGH, GHE, ABO, BGO, CDO and DAO i.e. 8 in number.

The triangles composed of eight components each are DAB, ABC, BCD and CDA i.e. 4 in number.

Total number of triangles in the figure = 16 + 16 + 8 + 4 = 44.

Squares :

The squares composed of two components are HIOL, IEJO, JFKO and KGLO i.e. 4 in number.

The squares composed of four components are AEOH, EBFO, OFGC and HOGD i.e.4 in number.

There is only one square EFGH which is composed of eight components.

There is only one square ABCD which is composed of sixteen components.

Total number of squares in the figure = 4 + 4 + 1 + 1 = 10.