The figure may be labeled as shown.
The simplest ||gms are ABFE, BCGF, CDHG, EFJI, FGKJ and GHLK. These are 6 in number.
The parallelograms composed of two components each are ACGE, BDHF, EGKI, FHLJ, ABJI, BCKJ and CDLK. Thus, there are 7 such parallelograms.
The parallelograms composed of three components each are ADHE and EHLI i.e. 2 in number.
The parallelograms composed of four components each are ACKI and BDLJ i.e. 2 in number
There is only one parallelogram composed of six components, namely ADLI.
Thus, there are 6 + 7 + 2 + 2 + 1 = 18 parallelograms in the figure.
In the adjoining figure, if the centres of all the circles are joined by horizontal and vertical lines, then find the number of squares that can be formed.
The figure may be labeled as shown.
We shall join the centres of all the circles by horizontal and vertical lines and then label the resulting figure as shown.
The simplest squares are ABED, BCFE, DEHG, EFIH, GHKJ and HILK i.e. 6 in number.
The squares composed of four simple squares are ACIG and DFLJ i.e. 2 in number.
Thus, 6 + 2 = 8 squares will be formed.
The figure may be labeled 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 labeled 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 labeled 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 labeled 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 GVWM i.e. 4 in number.
The rectangles composed of three components each are AQRP, PRMO, EGHT and THIK i.e. 4 in number.
The rectangles composed of four components each are CEKM, AVSP,TSWO, 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.
Answer & Explanation
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