Question 1

Find the number of triangles in the given figure.

Accepted Answer

The figure may be labelled as shown. The simplest triangles are AEI, AIH, BEJ, BJF, CFK, CKG, DGL, DLH, EOJ, FOJ, FOG, LOG, HOL and HOE i.e. 14 in number. The triangles composed of two components each are EAH, FBE, BEO, EOF, BFO, FCG, GDH, HOD, HOG and GOD i.e. 10 in number. The triangles composed of three components each are EFH, EHG, FGH and EFG i.e. 4 in number. Thus, there are 14 + 10 + 4 = 28 triangles in the given figure.

Question 2

Find the number of triangles in the given figure.

Accepted Answer

The figure may be labelled as shown. The simplest triangles are AHG, AIG, AIB, JFE, CJE and CED i.e. 6 in number. The triangles composed of two components each are ABG, CFE, ACJ and EGI i.e. 4 in number. The triangles composed of three components each are ACE, AGE and CFD i.e. 3 in number. There is only one triangle i.e. AHE composed of four components. Therefore, There are 6 + 4 + 3 + 1 = 14 triangles in the given figure.

Question 3

Count the number of parallelogram in the given figure.

Accepted Answer

The figure may be labelled as shown. The simplest parallelograms are EMLA and NIDJ i.e. 2 in number. The parallelograms composed of two components each are BFMG, CGNH, GMKN, FGME, GHNM, MNKL, FGNM, GHIN and MNJK i.e. 9 in number. The parallelograms composed of three components each are FGLA, ENKA, GHDJ and MIDK ie. 4 in number. The parallelograms composed of four components each are FGJK, GHKL, FBNK, CHKM, EFHN and MFHI i.e. 6 in number. The parallelograms composed of seven components each are FHKA and FHDK i.e. 2 in number. Total number of parallelograms in the figure = 2 + 9 + 4 + 6 + 2 = 23.

Question 4

Count the number of parallelogram in the given figure.

Accepted Answer

The figure may be labelled as shown. The parallelograms composed of two components each are ADME, DFNM, EMOG, FHJN, MNKO, GOLI, HBJN, NJKO, OKLI, FHNM, MNOG, DFME, HJKN, NKLO, OLCI, FNOM, MOIG and DMGE. i.e. 18 in number. The parallelograms composed of four components each are HOKB, NILJ, FGOH, HOLJ, NICK, FGIN, FMJB, DENH, MGKJ, MGCL, DEIO, FMLK, AENF, AGOD, DMJH, DOKF, EILM and EGKN i.e. 18 in number. The parallelograms composed of six components each are AEJH, DAIL, DECL, DEJB, HILB and HICJ i.e. 6 in number. The parallelograms composed of eight components each are FGKB, FGCK and AGKF i.e. 3 in number. Total number of parallelograms in the figure = 18 + 18 + 6 + 3 = 45.

Question 5

Count the number of rectangles in the given figure.

Accepted Answer

The figure may be labelled as shown. The rectangles composed of two components each are HIJE, EKJ,F, FMNG, GPQH, AEOH, EBFO, OFCG and HOGD i.e. 8 in number. The rectangles composed of four components each are ABFH, BCGE, CDHF, DAEG and EFGH i.e. 5 in number. The rectangles composed of six components each are IJFG, KLGH, MNHE and PQEF i.e. 4 in number. The rectangles composed of eight components each are IJMN, KLPQ and ABCD i.e. 3 in number. Thus, there are 8 + 5 + 4 + 3 = 20 rectangles in the given figure. (Here note that the squares are also counted amongst rectangles)

Question 6

Count the number of triangles and squares in the given figure.

Accepted Answer

The figure may be labelled as shown. Triangles: The simplest triangles are BGM, GHM, HAM, ABM, GIN, UN, JHN, HGN, IKO, KLO, LJO, JIO, KDP, DEP, ELP, LKP, BCD and AFE i.e. 18 in number. The triangles composed of two components each are ABG, BGH, GHA, HAB, HGI, GIJ, IJH, JHG, JIK, IKL, KLJ, LJI, LKD, KDE, DEL and ELK i.e. 16 in number. The triangles composed of four components each are BHI, GJK, ILD, AGJ, HIL and JKE i.e. 6 in number. Total number of triangles in the figure = 18 +16 + 6 = 40. Squares : The squares composed of two components each are MGNH, NIOJ and OKPL i.e. 3 in number. The squares composed of four components each are BGHA, GIJH, IKLJ and KDEL i.e. 4 in number. Total number of squares in the figure = 3 + 4 = 7.

Question 7

Count the number of triangles and squares in the given figure.

Accepted Answer

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.

Question 8

Count the number of squares in the given figure.

Accepted Answer

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.

Question 9

How many rectangles are there in the given figure.

Accepted Answer

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.

Question 10

Count the number of squares in the given figure.

Accepted Answer

The figure may be labelled as shown. The squares composed of two components each are BNQM, CORN, DPSO, MQTL, NRUQ, OSVR, PFWS, QUJT, RVIU and SWHV i.e. 10 in number. The squares composed of four components each are ABQL, BCRQ, CDSR, DEFS, LQJK, QRIJ, RSHI and SFGH i.e. 8 in number. The squares composed of eight components each are BRJL, CSIQ and DFHR i.e. 3 in number. The squares composed of sixteen components each are ACIK, BDHJ and CEGI i.e. 3 in number. Thus, there are 10 + 8 + 3 + 3 = 24 squares in the figure.

Question 11

What is the minimum number of straight lines that is needed to construct the figure?

Accepted Answer

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.

Question 12

Count the number of triangles and squares in the given figure.

Accepted Answer

The figure may be labelled as shown. Triangles : The simplest triangles are IJQ, JKQ, KLQ, LMQ, MNQ, NOQ, OPQ and PIQ i.e. 8 in number. The triangles composed of two components each are ABQ, BCQ, CDQ, DEQ, EFQ, FGQ, GHQ, HAQ, IKQ, KMQ, MOQ and OIQ i.e. 12 in number. The triangles composed of four components each are ACQ, CEQ, EGQ, GAQ, IKM, KMO, MOI and OIK i.e. 8 in number. The triangles composed of eight components each are ACE, CEG, EGA and GAC i.e. 4 in number. Total number of triangles in the figure = 8 + 12 + 8 + 4 = 32. Squares : The squares composed of two components each are IJQP, JKLQ, QLMN and PQNO i.e. 4 in number. The squares composed of four components each are ABQH, BCDQ, QDEF and HQFG i.e. 4 in number. There is only one square i.e. IKMO composed of eight components. There is only one square i.e. ACEG composed of sixteen components. Thus, there are 4 + 4 + 1 + 1= 10 squares in the given figure.

Question 13

Count the number of squares in the given figure.

Accepted Answer

The figure may be labelled as shown. The simplest squares are BCNM, CDON, PQIJ and QRHI i.e. 4 in number. The squares composed of two components each are MNTS, NOUT, STQP and TURQ i.e. 4 in number. The squares composed of five components each are ACTL, CEFT, TFGI and LTIK i.e. 4 in number. The squares composed of six components each are BDUS and SUHJ i.e. 2 in number. There is only one square i.e. MORP composed of eight components. There is only one square i.e. AEGK composed of twenty components. Total number of squares in the figure = 4 + 4 + 4 + 2+1 + 1 = 16.

Question 14

Count the number of rectangles in the given figure.

Accepted Answer

The figure may be labelled as shown. The simplest rectangles are ABQP, PQNO, BCDN, NDEM, MEFL, LFJK, FGHR and RHIJ i.e. 8 in number. The rectangles composed of two components each are ABNO, BCEM, NDFL, MEJK and FGIJ i.e. 5 in number. The rectangles composed of three components each are ACDO, BCFL, NDJK and LGIK i.e. 4 in number. There is only one rectangle i.e. BCJK composed of four components. Total number of rectangles in the figure = 8 + 5 + 4 + 1 = 18.

Question 15

Count the number of squares in the given figure.

Accepted Answer

The figure may be labelled as shown. The squares composed of two components each are BJMI, CKMJ, DLMK and AIML i.e. 4 in number. The squares composed of three components each are EBMA, BFCM, MCGD and AMDH i.e. 4 in number. The squares composed of four components each are VWBA, XYCB, ZA1DC and B1C1AD i.e. 4 in number. The squares composed of seven components each are NOJL, PQKI, RSLJ and TUIK i.e. 4 in number. There is only one square i.e. ABCD composed of eight components. There is only one square i.e. EFGH composed of twelve components. Total number of squares in the figure = 4 + 4 + 4 + 4 + 1 + 1 = 18.

Question 16

What is the minimum number of different colours required to paint he given figure such that no two adjacent regions have the same colour?

Accepted Answer

The figure may be labelled as shown. The regions A, C, E and G can have the same colour say colour 1. The regions B, D, F and H can have the same colour (but different from colour 1) say colour 2. The region 1 lies adjacent to each one of the regions A, B, C, D, E, F, G and H and therefore it should have a different colour say colour 3. The regions J, L and N can have the same colour (different from colour 3) say colour 1. The regions K, M and O can have the same colour (different fromthe colours 1 and 3). Thus, these regions will have colour 2. The region P cannot have any of the colours 1 and 2 as it lies adjacent to each one of the regions J, K, L, M, N and O and so it will have colour 3. The region Q can have any of the colours 1 or 2. Minimum number of colours required is 3.

Question 17

Count the number of triangles and squares in the given figure.

Accepted Answer

The figure may be labelled as shown. Triangles : The, simplest triangles are ABI, BGI, GHI, HAI, BCJ, CFJ, FGJ, GBJ, CDK, DEK, EFK and FCK i.e. 12 in number. The triangles composed of two components each are ABG, BGH, GHA, HAB, BCF, CFG, FGB, GBC, CDE, DEF, EFC and FGD i.e. 12 in number. The triangles composed of four components each are AGC, BFD, HBF and GCE i.e.4 in number. Thus, there are 12 + 12 + 4 = 28 triangles in the given figure. Squares : The squares composed of two components each are BJGI and CKFJ i.e. 2 in number. The squares composed of four components each are ABGH, BCFG and CDEF i.e. 3 in number. Total number of squares in the figure = 2 + 3 = 5.

Question 18

How many triangles and parallelograms are there in the following figure?

Accepted Answer

The figure may be labelled as shown. Triangles: The simplest triangles are KJN, KJO, CNB, OEF, JIL, JIM, BLA and MFG i.e. 8 in number. The triangles composed of two components each are CDJ, EDJ, NKO, JLM, JAH and JGH i.e. 6 in number. The triangles composed of three components each are BKI, FKI, CJA and EJG i.e. 4 in number. The triangles composed of four components each are CDE and AJG i.e. 2 in number. The only triangle composed of six components is BKF. Thus, there are 8 + 6 + 4 + 2 + 1 = 21 triangles in the given figure. Parallelograms : The simplest parallelograms are NJLB and JOFM i.e. 2 in number. The parallelograms composed of two components each are CDKB, DEFK, BIHA and IFGH i.e.4 in number. The parallelograms composed of three components each are BKJA, KFGJ, CJIB and JEFI i.e.4 in number. There is only one parallelogram i.e. BFGA composed of four components. The parallelograms composed of five components each are CDJA, DEGJ, CJHA and JEGH i.e.4 in number. The only parallelogram composed of six components is CEFB. The only parallelogram composed of ten components is CEGA. Thus, there are 2 + 4 + 4 + 1 + 4+ 1 + 1 = 17 parallelograms in the given figure. (Here note that the squares and rectangles are also counted amongst the parallelograms).

Question 19

How many circles are there in the adjoining figure.

Accepted Answer

The figure may be labelled as shown. There are 13 circles in the given figure. This is clear from the adjoining figure in which the centres of all the circles in the given figure have been numbered from 1 to 13.

Question 20

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.

Accepted Answer

The figure may be labelled 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.

Analytical Reasoning Questions