In ‘An
alytical Reasoning’, we analyse the problems related to the counting of some geometrical figures in a given complex figures. The objective can be met by a systematic analysis of the complex figure given.
The steps involved in the analysis of the complex figure would be clear from the following examples:
Analytical Reasoning Solved Example
Examples 1) Count the number of triangles in the problem figure.
Solution: As a first step, all the intersecting points in the problem figure are labeled as shown.
Now count the number of simplest triangles. These are ABD, BED, CBF, BEF, FIH, FHE, GDH, DHE and their number is 8.
Next count the triangles which are formed by the combination of two of these simple triangles. These are DBF, BFH, FHD and BHD and their number is 4.
Now since, the triangles formed with the combination of three or more triangles do not exist, the total number of triangles in the figure is 8 + 4 = 12. So (c) is the correct answer.
Example 2) Find the number of straight lines in the figure.
Solution: First the given problem figure is labeled as shown.
Now, count the horizontal lines, namely AC, HD and GE and their number is 3.
Now, count the vertical lines AG, BF and CE and their number is also 3.
Finally, count the six (6) slanting lines namely – BH, CG, DF, BD, AE and HF.
Thus, the total number of straight lines = 3 + 3 + 6 = 12
Hence, (a) is the answer.
Example 3) Count the number of squares in the given problem figure.
Solution: As usual we first label the given problem figure as shown. Now, the squares formed with two components are-namely, ABJI, BCKJ, CDLK, IJGH, JKFG and KLEF. Their number in 6.
Similarly, a square made of four components is only 1, i.e.., CJFL
Finally, the squares composed of eight components are ACFH and BDEG. So their number is 2.
∴ The total number of squares = 6 + 1 + 2 = 9.
So, (b) is the answer.
Example 4) Count the number of straight lines in the given problem figure.
Solution: The given problem figure may be labeled as shown:
Now we have
(i) Three (3) horizontal lines namely, AC, HD and GE.
(ii) Three (3) slanting lines namely, AG, BF and CE. And
(iii) Six (6) slanting lines namely, BH, CG, DF, BD, AE and HF.
∴ Total No. of lines = 3+3+6 = 12
∴ So (a) in the correct answer.
Example 5) Find the number of parallograms in the given problem figure.
Solution: First the given figure is labeled as shown.
Now,
(i) We have 6 simplest parallelograms namely, ABKJ, BCLK, CDEL, JKHI, KLOGH, LEFG.
(ii) The parallelograms composed of two components are ACLJ, BDEK, CKFG, EFHK; LGIJ, HIAB and, BCGH and their number is 7.
(iii) The parallogram composed of four components are ACGI and BDFH and their number is 2.
(iv) The parallogram composed of six elements is ADFI, i.e.., one only.
∴ The total number of parallelograms = 6+7+2+1 = 16
∴ The answer is (c).
Analytical Reasoning Questions from Previous Year Exam
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Question 1 of 45
1. Question
1 pointsFind the number of triangles in the given figure.
Correct
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.
Incorrect
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.
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Question 2 of 45
2. Question
1 pointsFind the minimum number of straight lines required to make the given figure.
Correct
The figure may be labelled as shown.
The horizontal lines are IK, AB, HG and DC i.e. 4 in number.The vertical lines are AD, EH, JM, FG and BC i.e. 5 in number.
The slanting lines are IE, JE, JF, KF, DE, DH, FC and GC i.e. 8 is number.
Thus, there are 4 + 5 + 8 = 17 straight lines in the figure.
Incorrect
The figure may be labelled as shown.
The horizontal lines are IK, AB, HG and DC i.e. 4 in number.The vertical lines are AD, EH, JM, FG and BC i.e. 5 in number.
The slanting lines are IE, JE, JF, KF, DE, DH, FC and GC i.e. 8 is number.
Thus, there are 4 + 5 + 8 = 17 straight lines in the figure.
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Question 3 of 45
3. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are AGH, GFO, LFO, DJK, EKP, PEL and IMN i.e. 7 in number.The triangles having two components each are GFL, KEL, AMO, NDP, BHN, CMJ, NEJ and HFM i.e. 8 in number.
The triangles having three components each are IOE, IFP, BIF and CEI i.e. 4 in number.
The triangles having four components each are ANE and DMF i.e. 2 in number.
The triangles having five components each are FCK, BGE and ADL i.e. 3 in number.
The triangles having six components each are BPF, COE, DHF and AJE i.e. 4 in number.
Total number of triangles in the figure = 7 + 8 + 4 + 2 + 3 + 4 = 28.
Incorrect
The figure may be labelled as shown.
The simplest triangles are AGH, GFO, LFO, DJK, EKP, PEL and IMN i.e. 7 in number.The triangles having two components each are GFL, KEL, AMO, NDP, BHN, CMJ, NEJ and HFM i.e. 8 in number.
The triangles having three components each are IOE, IFP, BIF and CEI i.e. 4 in number.
The triangles having four components each are ANE and DMF i.e. 2 in number.
The triangles having five components each are FCK, BGE and ADL i.e. 3 in number.
The triangles having six components each are BPF, COE, DHF and AJE i.e. 4 in number.
Total number of triangles in the figure = 7 + 8 + 4 + 2 + 3 + 4 = 28.
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Question 4 of 45
4. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are AHB, GHI, BJC, GFE, GIE, IJE, CEJ and CDE i.e. 8 in number.The triangles composed of two components each are HEG, BEC, HBE, JGE and ICE i.e. 5 in number.
The triangles composed of three components each are FHE, GCE and BED i.e. 3 in number.
There is only one triangle i.e. AGC composed of four components.
There is only one triangle i.e. AFD composed of nine components.
Thus, there are 8 + 5 + 3 + 1 + 1 = 18 triangles in the given figure.
Incorrect
The figure may be labelled as shown.
The simplest triangles are AHB, GHI, BJC, GFE, GIE, IJE, CEJ and CDE i.e. 8 in number.The triangles composed of two components each are HEG, BEC, HBE, JGE and ICE i.e. 5 in number.
The triangles composed of three components each are FHE, GCE and BED i.e. 3 in number.
There is only one triangle i.e. AGC composed of four components.
There is only one triangle i.e. AFD composed of nine components.
Thus, there are 8 + 5 + 3 + 1 + 1 = 18 triangles in the given figure.
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Question 5 of 45
5. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are IJO, BCJ, CDK, KQL, MLQ, GFM, GHN and NIO i.e. 8 in number.The triangles composed of two components each are ABO, AHO, NIJ, IGP, ICP, DEQ, FEQ, KLM, LCP and LGP i.e.10 in number.
The triangles composed of four components each are HAB, DEF, LGI, GIC, ICL and GLC i.e. 6 in number.
Total number of triangles in the figure = 8 + 10 + 6 = 24.
Incorrect
The figure may be labelled as shown.
The simplest triangles are IJO, BCJ, CDK, KQL, MLQ, GFM, GHN and NIO i.e. 8 in number.The triangles composed of two components each are ABO, AHO, NIJ, IGP, ICP, DEQ, FEQ, KLM, LCP and LGP i.e.10 in number.
The triangles composed of four components each are HAB, DEF, LGI, GIC, ICL and GLC i.e. 6 in number.
Total number of triangles in the figure = 8 + 10 + 6 = 24.
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Question 6 of 45
6. Question
1 pointsFind the minimum number of straight lines required to make the given figure.
Correct
The figure may be labelled as shown.
The horizontal lines are IJ, AB, EF, MN, HG, DC and LK i.e. 7 in number.The vertical lines are AD, EH, IL, FG, BC and JK i.e. 6 in number.
Thus, there are 7 + 6 = 13 straight lines in the figure.
Incorrect
The figure may be labelled as shown.
The horizontal lines are IJ, AB, EF, MN, HG, DC and LK i.e. 7 in number.The vertical lines are AD, EH, IL, FG, BC and JK i.e. 6 in number.
Thus, there are 7 + 6 = 13 straight lines in the figure.
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Question 7 of 45
7. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are AFJ, FJK, FKB, BKG, JKG, JGC, HJC, HIJ, DIH, DEI, EIJ and AEJ i.e. 12 in number.The triangles composed of two components each are JFB, FBG, BJG, JFG, DEJ, EJH, DJH and DEH i.e. 8 in number.
The triangles composed of three components each are AJB, JBC, DJC and ADJ i.e. 4 in number.
The triangles composed of six components each are DAB, ABC, BCD and ADC i.e. 4 in number.
Thus, there are 12 + 8 + 4 + 4 = 28 triangles in the figure.
Incorrect
The figure may be labelled as shown.
The simplest triangles are AFJ, FJK, FKB, BKG, JKG, JGC, HJC, HIJ, DIH, DEI, EIJ and AEJ i.e. 12 in number.The triangles composed of two components each are JFB, FBG, BJG, JFG, DEJ, EJH, DJH and DEH i.e. 8 in number.
The triangles composed of three components each are AJB, JBC, DJC and ADJ i.e. 4 in number.
The triangles composed of six components each are DAB, ABC, BCD and ADC i.e. 4 in number.
Thus, there are 12 + 8 + 4 + 4 = 28 triangles in the figure.
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Question 8 of 45
8. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are AKI, AIL, EKD, LFB, DJC, BJC, DHC and BCG i.e. 8 in number.The triangles composed of two components each are AKL, ADJ, AJB and DBC i.e. 4 in number.
The triangles composed of the three components each are ADC and ABC i.e. 2 in number.
There is only one triangle i.e. ADB composed of four components.
Thus, there are 8+ 4 + 2 + 1= 15 triangles in the figure.
Incorrect
The figure may be labelled as shown.
The simplest triangles are AKI, AIL, EKD, LFB, DJC, BJC, DHC and BCG i.e. 8 in number.The triangles composed of two components each are AKL, ADJ, AJB and DBC i.e. 4 in number.
The triangles composed of the three components each are ADC and ABC i.e. 2 in number.
There is only one triangle i.e. ADB composed of four components.
Thus, there are 8+ 4 + 2 + 1= 15 triangles in the figure.
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Question 9 of 45
9. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are ABF, BIC, CIH, GIH, FGE and AFE i.e. 6 in number.The triangles composed of two components each are ABE, AGE, BHF, BCH, CGH and BIE i.e. 6 in number.
The triangles composed of three components each are ABH, BCE and CDE i.e. 3 in number.
Hence, the total number of triangles in the figure = 6 + 6 + 3 = 15
Incorrect
The figure may be labelled as shown.
The simplest triangles are ABF, BIC, CIH, GIH, FGE and AFE i.e. 6 in number.The triangles composed of two components each are ABE, AGE, BHF, BCH, CGH and BIE i.e. 6 in number.
The triangles composed of three components each are ABH, BCE and CDE i.e. 3 in number.
Hence, the total number of triangles in the figure = 6 + 6 + 3 = 15
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Question 10 of 45
10. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are AGE, EGC, GFC, BGF, DGB and ADG i.e. 6 in number.The triangles composed of two components each are AGC, BGC and ABG i.e. 3 in number.
The triangles composed of three components each are AFC, BEC, BDC, ABF, ABE and DAC i.e. 6 in number.
There is only one triangle i.e. ABC composed of six components.
Thus, there are 6 + 3 + 6 + 1 = 16 triangles in the given figure.
Incorrect
The figure may be labelled as shown.
The simplest triangles are AGE, EGC, GFC, BGF, DGB and ADG i.e. 6 in number.The triangles composed of two components each are AGC, BGC and ABG i.e. 3 in number.
The triangles composed of three components each are AFC, BEC, BDC, ABF, ABE and DAC i.e. 6 in number.
There is only one triangle i.e. ABC composed of six components.
Thus, there are 6 + 3 + 6 + 1 = 16 triangles in the given figure.
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Question 11 of 45
11. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are ABL, BCD, DEF, FGP, PGH, QHI, JQI, KRJ and LRK i.e. 9 in number.The triangles composed of two components each are OSG, SGQ, SPI, SRI, KSQ, KMS, FGH, JHI and JKL i.e. 9 in number.
There is only one triangle i.e. KSG which is composed of four components.
The triangles composed of five components each are NEI, ANI, MCG and KCO i.e. 4 in number.
The triangles composed of six components each are GMK and KOG i.e. 2 in number.
There is only one triangle i.e. AEI composed of ten components.
There is only one triangle i.e. KCG composed of eleven components.
Therefore, Total number of triangles in the given figure = 9 + 9+1 + 4 + 2+1 + 1 = 27.
Incorrect
The figure may be labelled as shown.
The simplest triangles are ABL, BCD, DEF, FGP, PGH, QHI, JQI, KRJ and LRK i.e. 9 in number.The triangles composed of two components each are OSG, SGQ, SPI, SRI, KSQ, KMS, FGH, JHI and JKL i.e. 9 in number.
There is only one triangle i.e. KSG which is composed of four components.
The triangles composed of five components each are NEI, ANI, MCG and KCO i.e. 4 in number.
The triangles composed of six components each are GMK and KOG i.e. 2 in number.
There is only one triangle i.e. AEI composed of ten components.
There is only one triangle i.e. KCG composed of eleven components.
Therefore, Total number of triangles in the given figure = 9 + 9+1 + 4 + 2+1 + 1 = 27.
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Question 12 of 45
12. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are ABI, BIC, AIJ, CIJ, AHJ, CDJ, JHG, JDE, GJF and EJF i.e. 10 in number.The triangles composed of two components each are ABC, BCJ, ACJ, BAJ, AJG, CJE and GJE i.e. 7 in number.
The triangles composed of four components each are ACG, ACE, CGE and AGE i.e. 4 in number.
Total number of triangles in the figure =10+ 7 + 4 = 21.
Incorrect
The figure may be labelled as shown.
The simplest triangles are ABI, BIC, AIJ, CIJ, AHJ, CDJ, JHG, JDE, GJF and EJF i.e. 10 in number.The triangles composed of two components each are ABC, BCJ, ACJ, BAJ, AJG, CJE and GJE i.e. 7 in number.
The triangles composed of four components each are ACG, ACE, CGE and AGE i.e. 4 in number.
Total number of triangles in the figure =10+ 7 + 4 = 21.
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Question 13 of 45
13. Question
1 pointsFind the number of triangles in the given figure.
Correct
he figure may be labelled as shown.
The simplest triangles are AJF, FBG, GCH, HDI and IEJ i.e. 5 in number.The triangles composed of three components each EBH, AIC, EFC, ADG and BJD i.e. 5 in number.
Thus, there are 5 + 5 = 10 triangles in the figure.
Incorrect
he figure may be labelled as shown.
The simplest triangles are AJF, FBG, GCH, HDI and IEJ i.e. 5 in number.The triangles composed of three components each EBH, AIC, EFC, ADG and BJD i.e. 5 in number.
Thus, there are 5 + 5 = 10 triangles in the figure.
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Question 14 of 45
14. Question
1 pointsFind the minimum number of straight lines required to make the given figure.
Correct
The figure may be labelled as shown.
The horizontal lines are DE, FH, IL and BC i.e. 4 in number.The slanting lines are AC, DO, FN, IM, AB, EM and HN i.e. 7 in number.
Thus, there are 4 + 7 = 11 straight lines in the figure.
Incorrect
The figure may be labelled as shown.
The horizontal lines are DE, FH, IL and BC i.e. 4 in number.The slanting lines are AC, DO, FN, IM, AB, EM and HN i.e. 7 in number.
Thus, there are 4 + 7 = 11 straight lines in the figure.
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Question 15 of 45
15. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are ABJ, ACJ, BDH, DHF, CIE and GIE i.e. 6 in number.The triangles composed of two components each are ABC, BDF, CEG, BHJ, JHK, JKI and CJI i.e. 7 in number.
There is only one triangle JHI which is composed of four components.
Thus, there are 6 + 7 + 1 = 14 triangles in the given figure.
Incorrect
The figure may be labelled as shown.
The simplest triangles are ABJ, ACJ, BDH, DHF, CIE and GIE i.e. 6 in number.The triangles composed of two components each are ABC, BDF, CEG, BHJ, JHK, JKI and CJI i.e. 7 in number.
There is only one triangle JHI which is composed of four components.
Thus, there are 6 + 7 + 1 = 14 triangles in the given figure.
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Question 16 of 45
16. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are AHL, LHG, GHM, HMB, GMF, BMF, BIF, CIF, FNC, CNJ, FNE, NEJ, EKJ and JKD i.e. 14 in number.The triangles composed of two components each are AGH, BHG, HBF, BFG, HFG, BCF, CJF, CJE, JEF, CFE and JED i.e. 11 in number.
The triangles composed of four components each are ABG, CBG, BCE and CED i.e. 4 in number.
Total number of triangles in the given figure = 14 + 11 + 4 = 29.
Incorrect
The figure may be labelled as shown.
The simplest triangles are AHL, LHG, GHM, HMB, GMF, BMF, BIF, CIF, FNC, CNJ, FNE, NEJ, EKJ and JKD i.e. 14 in number.The triangles composed of two components each are AGH, BHG, HBF, BFG, HFG, BCF, CJF, CJE, JEF, CFE and JED i.e. 11 in number.
The triangles composed of four components each are ABG, CBG, BCE and CED i.e. 4 in number.
Total number of triangles in the given figure = 14 + 11 + 4 = 29.
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Question 17 of 45
17. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are APQ, AEQ, QTU, QRU, BGS, BHS, RSU, SUV, TUW, UWX, NWD, WDM, UVY, UXY, JCY and YKC i.e. 16 in number.The triangles composed of two components each are QUW, QSU, SYU and UWY i.e. 4 in number.
The triangles composed of three components each are AOU, AFU, FBU, BIU, UIC, ULC, ULD and OUD i.e. 8 in number.
The triangles composed of four components each are QYW, QSW, QSY and SYW i.e. 4 in number.
The triangles composed of six components each are AUD, ABU, BUC and DUC i.e. 4 in number.
The triangles composed of seven components each are QMC, ANY, EBW, PSD, CQH, AGY, DSK and BJW i.e. 8 in number.
The triangles composed of twelve components each are ABD, ABC, BCD and ACD i.e. 4 in number.
Thus, there are 16 + 4 + 8 + 4 + 4 + 8 + 4 = 48 triangles in the figure.
Incorrect
The figure may be labelled as shown.
The simplest triangles are APQ, AEQ, QTU, QRU, BGS, BHS, RSU, SUV, TUW, UWX, NWD, WDM, UVY, UXY, JCY and YKC i.e. 16 in number.The triangles composed of two components each are QUW, QSU, SYU and UWY i.e. 4 in number.
The triangles composed of three components each are AOU, AFU, FBU, BIU, UIC, ULC, ULD and OUD i.e. 8 in number.
The triangles composed of four components each are QYW, QSW, QSY and SYW i.e. 4 in number.
The triangles composed of six components each are AUD, ABU, BUC and DUC i.e. 4 in number.
The triangles composed of seven components each are QMC, ANY, EBW, PSD, CQH, AGY, DSK and BJW i.e. 8 in number.
The triangles composed of twelve components each are ABD, ABC, BCD and ACD i.e. 4 in number.
Thus, there are 16 + 4 + 8 + 4 + 4 + 8 + 4 = 48 triangles in the figure.
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Question 18 of 45
18. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are ABF, BFG, BCG, CGH, GHD, GED, EFG and AFE i.e. 8 in number.The triangles composed of two components each are ABG, BGE, AGE, ABE and GCD i.e. 5 in number.
The triangles composed of three components each are BCD, CDE, BED and BCE i.e. 4 in number.
Thus, there are 8 + 5 + 4 = 17 triangles in the figure.
Incorrect
The figure may be labelled as shown.
The simplest triangles are ABF, BFG, BCG, CGH, GHD, GED, EFG and AFE i.e. 8 in number.The triangles composed of two components each are ABG, BGE, AGE, ABE and GCD i.e. 5 in number.
The triangles composed of three components each are BCD, CDE, BED and BCE i.e. 4 in number.
Thus, there are 8 + 5 + 4 = 17 triangles in the figure.
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Question 19 of 45
19. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are AEH, EHI, EBF, EFI, FGC, IFG, DGH and HIG i.e. 8 in number.The triangles composed of two components each are HEF, EFG, HFG and EFG i.e. 4 in number.
Thus, there are 8 + 4 = 12 triangles in the figure.
Incorrect
The figure may be labelled as shown.
The simplest triangles are AEH, EHI, EBF, EFI, FGC, IFG, DGH and HIG i.e. 8 in number.The triangles composed of two components each are HEF, EFG, HFG and EFG i.e. 4 in number.
Thus, there are 8 + 4 = 12 triangles in the figure.
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Question 20 of 45
20. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are ABG, BCG, CGE, CDE, AGE and AEF i.e. 6 in number.The triangles composed of two components each are ABE, ABC, BCE and ACE i.e. 4 in number.
There are 6 + 4 = 10 triangles in the figure.
Incorrect
The figure may be labelled as shown.
The simplest triangles are ABG, BCG, CGE, CDE, AGE and AEF i.e. 6 in number.The triangles composed of two components each are ABE, ABC, BCE and ACE i.e. 4 in number.
There are 6 + 4 = 10 triangles in the figure.
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Question 21 of 45
21. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are BFG, CGH, EFM, FMG, GMN, GHN, HNI, LMK, MNK and KNJ i.e. 10 in number.The triangles composed of three components each are FAK and HKD i.e. 2 in number.
The triangles composed of four components each are BEN, CMI, GLJ and FHK i.e. 4 in number.
The triangles composed of eight components each are BAJ and OLD i.e. 2 in number.
Thus, there are 10 + 2 + 4 + 2 = 18 triangles in the given figure.
Incorrect
The figure may be labelled as shown.
The simplest triangles are BFG, CGH, EFM, FMG, GMN, GHN, HNI, LMK, MNK and KNJ i.e. 10 in number.The triangles composed of three components each are FAK and HKD i.e. 2 in number.
The triangles composed of four components each are BEN, CMI, GLJ and FHK i.e. 4 in number.
The triangles composed of eight components each are BAJ and OLD i.e. 2 in number.
Thus, there are 10 + 2 + 4 + 2 = 18 triangles in the given figure.
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Question 22 of 45
22. Question
1 pointsFind the number of triangles in the given figure.
Correct
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.
Incorrect
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.
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Question 23 of 45
23. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are ABG, BIG, BIC, CIH, GIH, CDH, HED, GHJ, HJE, FEJ, GFJ and AGF i.e. 12 in number.The triangles composed of two components each are ABF, CDE, GBC, BCH, GHG, BHG, GHF, GHE, HEF and GEF i.e. 10 in number.
The triangles composed of three components each are ABH, AFH, CDG and GDE i.e. 4 in number.
The triangles composed of four components each are BHF and CGE i.e. 2 in number.
Total number of triangles in the figure = 12 + 10 + 4 + 2 = 28.
Incorrect
The figure may be labelled as shown.
The simplest triangles are ABG, BIG, BIC, CIH, GIH, CDH, HED, GHJ, HJE, FEJ, GFJ and AGF i.e. 12 in number.The triangles composed of two components each are ABF, CDE, GBC, BCH, GHG, BHG, GHF, GHE, HEF and GEF i.e. 10 in number.
The triangles composed of three components each are ABH, AFH, CDG and GDE i.e. 4 in number.
The triangles composed of four components each are BHF and CGE i.e. 2 in number.
Total number of triangles in the figure = 12 + 10 + 4 + 2 = 28.
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Question 24 of 45
24. Question
1 pointsFind the minimum number of straight lines required to make the given figure.
Correct
The figure may be labelled as shown.
The horizontal lines are AK, BJ, CI, DH and EG i.e. 5 in number.The vertical lines are AE, LF and KG i.e. 3 in number.
The slanting lines are LC, CF, FI, LI, EK and AG i.e. 6 in number.
Thus, there are 5 + 3 + 6 = 14 straight lines in the figure.
Incorrect
The figure may be labelled as shown.
The horizontal lines are AK, BJ, CI, DH and EG i.e. 5 in number.The vertical lines are AE, LF and KG i.e. 3 in number.
The slanting lines are LC, CF, FI, LI, EK and AG i.e. 6 in number.
Thus, there are 5 + 3 + 6 = 14 straight lines in the figure.
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Question 25 of 45
25. Question
1 pointsWhat is the number of straight lines and the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The Horizontal lines are DF and BC i.e. 2 in number.The Vertical lines are DG, AH and FI i.e. 3 in number.
The Slanting lines are AB, AC, BF and DC i.e. 4 in number.
Thus, there are 2 + 3 + 4 = 9 straight lines in the figure.
Now, we shall count the number of triangles in the figure.
The simplest triangles are ADE, AEF, DEK, EFK, DJK, FLK, DJB, FLC, BJG and LIC i.e. 10 in number.
The triangles composed of two components each are ADF, AFK, DFK, ADK, DKB, FCK, BKH, KHC, DGB and FIC i.e. 10 in number.
The triangles composed of three components each are DFJ and DFL i.e. 2 in number.
The triangles composed of four components each are ABK, ACK, BFI, CDG, DFB, DFC and BKC i.e. 7 in number.
The triangles composed of six components each are ABH, ACH, ABF, ACD, BFC and CDB i.e. 6 in number.
There is only one triangle i.e. ABC composed of twelve components.
There are 10 + 10 + 2 + 7 + 6+ 1 = 36 triangles in the figure.
Incorrect
The figure may be labelled as shown.
The Horizontal lines are DF and BC i.e. 2 in number.The Vertical lines are DG, AH and FI i.e. 3 in number.
The Slanting lines are AB, AC, BF and DC i.e. 4 in number.
Thus, there are 2 + 3 + 4 = 9 straight lines in the figure.
Now, we shall count the number of triangles in the figure.
The simplest triangles are ADE, AEF, DEK, EFK, DJK, FLK, DJB, FLC, BJG and LIC i.e. 10 in number.
The triangles composed of two components each are ADF, AFK, DFK, ADK, DKB, FCK, BKH, KHC, DGB and FIC i.e. 10 in number.
The triangles composed of three components each are DFJ and DFL i.e. 2 in number.
The triangles composed of four components each are ABK, ACK, BFI, CDG, DFB, DFC and BKC i.e. 7 in number.
The triangles composed of six components each are ABH, ACH, ABF, ACD, BFC and CDB i.e. 6 in number.
There is only one triangle i.e. ABC composed of twelve components.
There are 10 + 10 + 2 + 7 + 6+ 1 = 36 triangles in the figure.
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Question 26 of 45
26. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are AML, LRK, KWD, DWJ, JXI, IYC, CYH, HTG, GOB, BOF, FNE and EMA i.e. 12 in number.The triangles composed of two components each are AEL, KDJ, HIC and FBG i.e. 4 in number.
The triangles composed of three components each are APF, EQB, BQH, GVC, CVJ, IUD, DUL and KPA i.e. 8 in number.
The triangles composed of six components each are ASB, BSG, CSD, DSA, AKF, EBH, GGJ and IDL i.e. 8 in number.
The triangles composed of twelve components each are ADB, ABC, BCD and CDA i.e. 4 in number.
Total number of triangles in the figure = 12 + 4 + 8 + 8 + 4 = 36.
Incorrect
The figure may be labelled as shown.
The simplest triangles are AML, LRK, KWD, DWJ, JXI, IYC, CYH, HTG, GOB, BOF, FNE and EMA i.e. 12 in number.The triangles composed of two components each are AEL, KDJ, HIC and FBG i.e. 4 in number.
The triangles composed of three components each are APF, EQB, BQH, GVC, CVJ, IUD, DUL and KPA i.e. 8 in number.
The triangles composed of six components each are ASB, BSG, CSD, DSA, AKF, EBH, GGJ and IDL i.e. 8 in number.
The triangles composed of twelve components each are ADB, ABC, BCD and CDA i.e. 4 in number.
Total number of triangles in the figure = 12 + 4 + 8 + 8 + 4 = 36.
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Question 27 of 45
27. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are GLK, DLJ, DJM, HMN, QRE, IRA, IPA and FPO i.e. 8 in number.The triangles having two components each are BDO, CDQ, DLM, PRA, KFI, NEI, HJI, GJI, DKI and DNI i.e. 10 in number.
The triangles having four components each are DIE, DFI, DOA, DQA andGHI i.e. 5 in number.
The triangles having six components each are DCA and DBA i.e. 2 in number.
DEF is the only triangle having eight components.
ABC is the only triangle having twelve components.
Thus, there are 8+10+ 5 + 2+1 + 1 = 27 triangles in the figure.
Incorrect
The figure may be labelled as shown.
The simplest triangles are GLK, DLJ, DJM, HMN, QRE, IRA, IPA and FPO i.e. 8 in number.The triangles having two components each are BDO, CDQ, DLM, PRA, KFI, NEI, HJI, GJI, DKI and DNI i.e. 10 in number.
The triangles having four components each are DIE, DFI, DOA, DQA andGHI i.e. 5 in number.
The triangles having six components each are DCA and DBA i.e. 2 in number.
DEF is the only triangle having eight components.
ABC is the only triangle having twelve components.
Thus, there are 8+10+ 5 + 2+1 + 1 = 27 triangles in the figure.
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Question 28 of 45
28. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are ADE, BDF, DEF and EFC i.e. 4 in number.There is only one triangle ABC composed of four components.
Thus, there are 4+1 = 5 triangles in the given figure.
Incorrect
The figure may be labelled as shown.
The simplest triangles are ADE, BDF, DEF and EFC i.e. 4 in number.There is only one triangle ABC composed of four components.
Thus, there are 4+1 = 5 triangles in the given figure.
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Question 29 of 45
29. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are EFH, BIC, GHJ, GIJ, EKD and CKD i.e. 6 in number.The triangles composed of two components each are ABJ, AFJ, GCK, GEK, CED arid GHI i.e. 6 in number.
The triangles composed of three components each are GCD, GED, DJB and DJF i.e. 4 in number.
The triangles composed of four components each are ABF and GCE i.e. 2 in number.
The triangles composed of five components each are ABD and AFD i.e. 2 in number.
There is only one triangle i.e. FBD composed of six components.
Total number of triangles in the figure = 6 + 6 + 4 + 2 + 2 + 1 = 21.
Incorrect
The figure may be labelled as shown.
The simplest triangles are EFH, BIC, GHJ, GIJ, EKD and CKD i.e. 6 in number.The triangles composed of two components each are ABJ, AFJ, GCK, GEK, CED arid GHI i.e. 6 in number.
The triangles composed of three components each are GCD, GED, DJB and DJF i.e. 4 in number.
The triangles composed of four components each are ABF and GCE i.e. 2 in number.
The triangles composed of five components each are ABD and AFD i.e. 2 in number.
There is only one triangle i.e. FBD composed of six components.
Total number of triangles in the figure = 6 + 6 + 4 + 2 + 2 + 1 = 21.
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Question 30 of 45
30. Question
1 pointsFind the number of triangles in the given figure.
Correct
The figure may be labelled as shown.
The simplest triangles are AFB, FEB, EBC, DEC, DFE and AFD i.e. 6 in number.The triangles composed of two components each are AEB, FBC, DFC, ADE, DBE and ABD i.e. 6 in number.
The triangles composed of three components each are ADC and ABC i.e. 2 in number.
There is only one triangle i.e. DBC which is composed of four components.
Thus, there are 6 + 6 + 2 + 1 = 15 triangles in the figure.
Incorrect
The figure may be labelled as shown.
The simplest triangles are AFB, FEB, EBC, DEC, DFE and AFD i.e. 6 in number.The triangles composed of two components each are AEB, FBC, DFC, ADE, DBE and ABD i.e. 6 in number.
The triangles composed of three components each are ADC and ABC i.e. 2 in number.
There is only one triangle i.e. DBC which is composed of four components.
Thus, there are 6 + 6 + 2 + 1 = 15 triangles in the figure.
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Question 31 of 45
31. Question
1 pointsCount the number of squares in the given figure.
Correct
The figure may be labelled as shown.
The simplest squares are ABGF, BCHG, CDIH, DEJI, FGLK, GHML, HINM, IJON, KLQP, LMRQ, MNSR, NOTS, PQVU, QRWV, RSXW and STYX i.e. 16 in number.The squares composed of four components each are ACMK, BDNL, CEOM, FHRP, GISQ, HJTR, KMWU, LNXV and MOYW i.e. 9 in number.
The squares composed of nine components each are ADSP, BETQ, FIXU and GJYV i.e. 4 in number.
There is one square AEYU composed of sixteen components.
There are 16 + 9 + 4 + 1 = 30 squares in the given figure.
Incorrect
The figure may be labelled as shown.
The simplest squares are ABGF, BCHG, CDIH, DEJI, FGLK, GHML, HINM, IJON, KLQP, LMRQ, MNSR, NOTS, PQVU, QRWV, RSXW and STYX i.e. 16 in number.The squares composed of four components each are ACMK, BDNL, CEOM, FHRP, GISQ, HJTR, KMWU, LNXV and MOYW i.e. 9 in number.
The squares composed of nine components each are ADSP, BETQ, FIXU and GJYV i.e. 4 in number.
There is one square AEYU composed of sixteen components.
There are 16 + 9 + 4 + 1 = 30 squares in the given figure.
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Question 32 of 45
32. Question
1 pointsFind the number of quadrilaterals in the given figure.
Correct
The figure may be labelled as shown.
The quadrilaterals in the figure are ABCD, ABDE, ABDF, ABDH, CDHA, CDEA, CDFA, DEAG, DEFA, FAGD and AGDH.The number of quadrilaterals in the figure is 11.
Incorrect
The figure may be labelled as shown.
The quadrilaterals in the figure are ABCD, ABDE, ABDF, ABDH, CDHA, CDEA, CDFA, DEAG, DEFA, FAGD and AGDH.The number of quadrilaterals in the figure is 11.
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Question 33 of 45
33. Question
1 pointsCount the number of squares in the given figure.
Correct
The figure may be labelled as shown.
The simplest squares are QUYX, URVY, YVSW and XYWT i.e. 4 in number.The squares composed of two components each are IMYP, MJNY, YNKO and PYOL i.e. 4 in number.
The squares composed of three components each are AEYH, EBFY, YFCG and HYGD i.e. 4 in number.
There is only one square i.e. QRST composed of four components.
There is only one square i.e. IJKL composed of eight components.
There is only one square i.e. ABCD composed of twelve components.
Total number of squares in the given figure = 4 + 4 + 4+1 + 1 + 1 = 15.
Incorrect
The figure may be labelled as shown.
The simplest squares are QUYX, URVY, YVSW and XYWT i.e. 4 in number.The squares composed of two components each are IMYP, MJNY, YNKO and PYOL i.e. 4 in number.
The squares composed of three components each are AEYH, EBFY, YFCG and HYGD i.e. 4 in number.
There is only one square i.e. QRST composed of four components.
There is only one square i.e. IJKL composed of eight components.
There is only one square i.e. ABCD composed of twelve components.
Total number of squares in the given figure = 4 + 4 + 4+1 + 1 + 1 = 15.
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Question 34 of 45
34. Question
1 pointsWhat is the minimum number of colours required to fill the spaces in the given diagram without any two adjacent spaces having the same colour?
Correct
The figure may be labelled as shown.
The spaces P, Q and R have to be shaded by three different colours definitely (since each of these three spaces lies adjacent to the other two).Now, in order that no two adjacent spaces be shaded by the same colour, the spaces T, U and S must be shaded with the colours of the spaces P, Q and R respectively.
Also the spaces X, V and W must be shaded with the colours of the spaces S, T and U respectively i.e. with the colours of the spaces R, P and Q respectively. Thus, minimum three colours are required.
Incorrect
The figure may be labelled as shown.
The spaces P, Q and R have to be shaded by three different colours definitely (since each of these three spaces lies adjacent to the other two).Now, in order that no two adjacent spaces be shaded by the same colour, the spaces T, U and S must be shaded with the colours of the spaces P, Q and R respectively.
Also the spaces X, V and W must be shaded with the colours of the spaces S, T and U respectively i.e. with the colours of the spaces R, P and Q respectively. Thus, minimum three colours are required.
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Question 35 of 45
35. Question
1 pointsCount the number of triangles and squares in the given figure.
Correct
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.
Incorrect
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.
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Question 36 of 45
36. Question
1 pointsCount the number of triangles and squares in the given figure.
Correct
The figure may be labelled as shown.
Triangles:The simplest triangles are JBO, BKO, KDO, DFO, FGO, GHO, HIO, IJO, ABJ, BCK, CKD and DEF i.e.12 in number.
The triangles composed of two components each are IBO, BDO, DGO, GIO, ABO, CDO, CBO, CBD and DEO i.e. 9 in number.
The triangles composed of four components each are IBD, BDG, DGI, GIB, ACO and COE i.e. 6 in number.
There is only one. triangle i.e. ACE composed of eight components.
Thus, there are 12 + 9 + 6 + 1 = 28 triangles in the given figure.
Squares:
The squares composed of two components each are BKOJ, KDFO, OFGH and JOHI i.e. 4 in number.
There is only one square i.e. CDOB composed of four components.
There is only one square i.e. BDGI composed of eight components.
Thus, there are 4 + 1 + 1 = 6 squares in the given figure.
Incorrect
The figure may be labelled as shown.
Triangles:The simplest triangles are JBO, BKO, KDO, DFO, FGO, GHO, HIO, IJO, ABJ, BCK, CKD and DEF i.e.12 in number.
The triangles composed of two components each are IBO, BDO, DGO, GIO, ABO, CDO, CBO, CBD and DEO i.e. 9 in number.
The triangles composed of four components each are IBD, BDG, DGI, GIB, ACO and COE i.e. 6 in number.
There is only one. triangle i.e. ACE composed of eight components.
Thus, there are 12 + 9 + 6 + 1 = 28 triangles in the given figure.
Squares:
The squares composed of two components each are BKOJ, KDFO, OFGH and JOHI i.e. 4 in number.
There is only one square i.e. CDOB composed of four components.
There is only one square i.e. BDGI composed of eight components.
Thus, there are 4 + 1 + 1 = 6 squares in the given figure.
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Question 37 of 45
37. Question
1 pointsCount the number of parallelogram in the given figure.
Correct
The figure may be labelled 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.
Incorrect
The figure may be labelled 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.
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Question 38 of 45
38. Question
1 pointsCount the number of triangles and squares in the given figure.
Correct
The figure may be labelled as shown.
Triangles :The simplest triangles are BPN, PNE, ABM, EFG, MLK, GHI, QRO, RSO, STO and QTO i.e. 10 in number.
The triangles composed of two components each are BPE, TQR, QRS, RST and STQ i.e. 5 in number.
The triangles composed of three components each are MPO and GPO i.e. 2 in number.
The triangles composed of six components each are LPJ, HPJ and MPG i.e. 3 in number.
There is only one triangle LPH composed of twelve components.
Total number of triangles in the figure = 10 + 5 - 2 + 3 + l = 21.
Squares :
The squares composed of two components each are KJOM and JIGQ i.e. 2 in number.
The squares composed of three components each are ANOM, NFGO and CDEB i.e.3 in number.
There is only one square i.e. QRST composed of four components.
There is only one square i.e. AFIK composed of ten components.
Total number of squares in the figure = 2 + 3 + 1 + 1 = 7.
Incorrect
The figure may be labelled as shown.
Triangles :The simplest triangles are BPN, PNE, ABM, EFG, MLK, GHI, QRO, RSO, STO and QTO i.e. 10 in number.
The triangles composed of two components each are BPE, TQR, QRS, RST and STQ i.e. 5 in number.
The triangles composed of three components each are MPO and GPO i.e. 2 in number.
The triangles composed of six components each are LPJ, HPJ and MPG i.e. 3 in number.
There is only one triangle LPH composed of twelve components.
Total number of triangles in the figure = 10 + 5 - 2 + 3 + l = 21.
Squares :
The squares composed of two components each are KJOM and JIGQ i.e. 2 in number.
The squares composed of three components each are ANOM, NFGO and CDEB i.e.3 in number.
There is only one square i.e. QRST composed of four components.
There is only one square i.e. AFIK composed of ten components.
Total number of squares in the figure = 2 + 3 + 1 + 1 = 7.
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Question 39 of 45
39. Question
1 pointsHow many triangles and parallelograms are there in the following figure?
Correct
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).
Incorrect
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).
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Question 40 of 45
40. Question
1 pointsCount the number of squares in the given figure.
Correct
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.
Incorrect
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.
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Question 41 of 45
41. Question
1 pointsCount the number of rectangles in the given figure.
Correct
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)
Incorrect
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)
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Question 42 of 45
42. Question
1 pointsWhat is the minimum number of straight lines that is needed to construct the figure?
Correct
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.
Incorrect
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.
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Question 43 of 45
43. Question
1 pointsDetermine the number of rectangles and hexagons in the given figure.
Correct
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 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.
Incorrect
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 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.
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Question 44 of 45
44. Question
1 pointsCount the number of rectangles in the given figure.
Correct
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.
Incorrect
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.
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Question 45 of 45
45. Question
1 pointsCount the number of squares in the given figure.
Correct
The figure may be labelled as shown.
The squares composed of two components each are ABKJ, BCLK, CDEL, LEFG, KLGH and JKHI i.e.6 in number.
There is only one square i.e. CEGK composed of four components.
The squares composed of eight components each are ACGI and BDFH i.e. 2 in number.
There are 6 + 1 + 2 = 9 squares in the figure.
Incorrect
The figure may be labelled as shown.
The squares composed of two components each are ABKJ, BCLK, CDEL, LEFG, KLGH and JKHI i.e.6 in number.
There is only one square i.e. CEGK composed of four components.
The squares composed of eight components each are ACGI and BDFH i.e. 2 in number.
There are 6 + 1 + 2 = 9 squares in the figure.
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