After labelling the figure, we get the following,

• Simplest triangles AEI, EOJ, OHI, HAI, EBJ, BFJ, FOJ, OEJ, HOL, OGL, GDL, DHL, OFK, FCK,CGK, and GOK (16)
• Triangles with two components each HAE, AEO, EOH, OHA, OEB,EBF, BFO, FOE, DHO, HOG, OGD, GDH, GOF, OFC, FCG, and CGO (16)
• Triangles with four components each HEF, EFG, FGH, GHE, ABO, BCO, CDO, and DAO (8)
• Triangles with eight components each DAB, ABC, BCD and CDA (4)

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

Also we need to count the number of squares in this figure,

• Squares with two components each HIOL, IEJO, JFKO, and KGLO (4)
• Squares with four components each AEOH, EBFO, OFGC and HOGD (4)
• Square with eight components is EFGH (1)
• Square with sixteen components is ABCD (1)

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

Hence option A is correct.