Notes - Diagrammatic Puzzles
Category : 12th Class
Diagrammatic Puzzles
In these problems one has to count the geometrical figures in a given complex figure. A little bit of systematic approach is needed to get the correct number of the asked figure. The shape of geometrical figures must be clear in mind.
Example:
1. How many triangles are there in the figure given below?
(a) 11 (b) 12
(c) 9 (d) 10
Ans. (b)
Explanation: The main triangle is: ABC - i.e. –1 triangle the simplest triangles are: ABE, AED, EBF, EDC and EFC i.e. -5 triangles
The triangle divided into two parts are: ABF, ABD, AEC and EBC i.e. - 4 triangles
Other triangles are: AFC and DBC i.e. - 2 triangles
So, the total number of triangles is \[1+5+4+2=12\]
2. The maximum number of squares in the figure is
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(a) 9 (b) 10
(c) 13 (d) 14
Ans. (d)
Explanation: The main square is ABCD i.e. - 1 square. The simplest squares are AMEF, MNFG, NBGH, EFU,
FGJK, GHKL, IJCO, JKOP and KLDP i.e. - 9 squares
Other squares are: ANIK, MBJL, EGCP and FHOD i.e. - 4 squares
So, the total number of squares is \[1+9+4=14\]
3. The number of rectangles in this figure are.
(a) 21 (b) 24
(c) 23 (d) 25
Ans. (c)
Explanation: The main rectangle is ABCD - i.e., -1 rectangle
The simplest rectangles are AIEJ, ILJM, LBMF, EJGK, JMKN, MFNH, GKCO and KHOD i.e. - 8 rectangles.
The rectangles which have two parts are ALEM, IBJF, EMGN, JFKH, AIGK, IKLN, LBNH, EJCO and GHCD i.e.,
-9 rectangles.
The rectangles which have three parts are AICO,
ABEF and EFGH i.e. - 3 rectangles
The rectangles which have four parts are ALGN and IBKH i.e. - 2 rectangles \[1+8+9+3+2=23\]
4. How many number of parallelograms in this figure?
(a) 11 (b) 12
(c) 9 (d) 10
Ans. (a)
Explanation: The main parallelogram in ABCD - i.e. –1 parallelogram
The simplest parallelogram are: AEFG, FGCI, EBGH, GHID, FECG and GBIH i.e., - 6 parallelograms
Other parallelograms are: ABFH, FHCD, AECI and EBID
i.e., - 4 parallelograms
So, the total number of parallelograms is \[1+6+4=11\]
5. How many circles are there in the figure given below?
(a) 5 (b) 6
(c) 8 (d) 9
Ans. (d)
Explanation: There is 1 main circle, 4 circles on the top of the horizontal line and 4 circles below it.
So, the total number of circles is \[1+4+4=9\]
Snap Test
1. How many triangles are there in the figure given below?
(a) 24 (b) 27
(c) 25 (d) 26
Ans. (b)
Explanation
2. How many triangles are there in the given figure?
(a) 15 (b) 14
(c) 12 (d) 13
Ans. (a)
Explanation: The simplest triangles are: ADE, ABC, BCG, CFG, EFH and DEH i.e. - 6 triangles
The triangles divided into two parts are: ADH, AEG, ABH, AEG, BFG and DFH i.e. - 6 triangles.
The triangles divided into three parts are: ADG, ABH and BHI i.e. - 3 triangles
So, the total number of triangles is \[6+6+3=15\]
3. How many squares are there in the figure?
(a) 36 (b) 60
(c) 77 (d) 91
Ans. (d)
Explanation: In such figures where the number of squares in rows and columns contained inside one square is equal, the manner of calculating the total number of squares is:
\[1\times 1=1\]
\[2\times 2=5\text{ }i.e.\left( 2\times 2 \right)+1\]
\[3\times 3=14\text{ }\,\,i.e.\text{ }\left( 3\times 3 \right)+5\]
\[4\times 4=30\,\,\,~i.e.\text{ }\left( 4\times 4 \right)+14\]
\[5\times 5=55\text{ }i.e.\text{ }\left( 5\times 5 \right)+30\]
\[6\times 6=91\text{ }\,\,i.e.\text{ }\left( 6\times 6 \right)+55\]
\[7\times 7=\left( 7\times 7 \right)+55=104\] and so on.
In this case the squares are \[6\times 6\] and so the total number of squares are 91.
4. How many number of triangles in the figure given below?
(a) 28 (b) 30
(c) 39 (d) 32
Ans. (c)
Explanation: The simplest triangles are: ABD, BEF, BCF, CFG, CGH, CDH, DHI, FJL, FJG, JLM, JGM, GKM, GKH, KMN, KHN, LOP, LMP, MPN and NPQ i.e. - 19 triangles
The bisected triangles are: CFH, FGM, FLM, FLG, GLM, GMN, GHN, GMH, HMN, LOP, LMP, MNP and PNQ i.e. - 13 triangles.
Other triangles are: AEI, GLN, FHM, CFM, CHM, GLP and GNP i.e. -7 triangles total triangles is \[19+13+7=39\]
5. How many sets of parallel lines are there in the figure given below?
(a) 8 (b) 7
(c) 9 (d) 10
Ans. (c)
Explanation: The horizontal lines are AB, CD, EF and GH i.e. - 4 lines
The vertical lines are AG, LM and BH i.e. - 3 lines
So, the total number of lines used to draw this figure is \[4+3+2=9\]
6. How many straight lines are used to given this figure?
(a) 7 (b) 11
(c) 8 (d) 10
Ans. (b)
Explanation: The horizontal lines are AB, CD, EF, GH and IJ i.e. - 5 lines
The vertical lines are AC, KL, MJ, BD, FH and GI i.e. - 6 lines
So, the total number of lines used to draw this figure is \[5+6=11\]
7. How many parallelograms are there in this figure?
(a) 49 (b) 46
(c) 44 (d) 39
Ans. (d)
Explanation: The main parallelograms are: ABCD and EFGH i.e. - 2 parallelograms
The simplest parallelograms are: AEIM, EKMN, KLNO, LFOP, FBPJ, JMCQ, MNQR, NORS, OPST, PJTD, QRGU, RSUV and STVH i.e. - 13 parallelograms.
The parallelogram divided into two parts are: AKIN, ELMO, KFNP, LBOJ, INCR, MOQS, NPRT, OJSD, QSGV, RTUH, AECQ, EKQR, MNGU, KLRS, NOUV, LFST, OPVH and FBTD i.e. -18 parallelograms.
The parallelograms divided into three parts are: ALIO, EFMP, KBNJ, IOCS, MPQT, NJRD, QTGH, EKGU, KLUV and LFVH i.e. - 10 parallelograms.
The parallelograms divided into four parts are: AFIF, EBQD, IPCT and MJQD i.e. – 4 parallelograms.
Other parallelograms are: ABIJ and IJCD i.e. - 2 parallelograms
So, the total number of parallelograms are: \[2+13+18+10+4+2=49\]
8. How many triangles are there in the figure given below?
(a) 19 (b) 21
(c) 20 (d) 19
Ans. (b)
Explanation: The main triangles are ABC and DEF i.e. - 2 triangles
The simplest triangles are: ADG, DGH, DHI, DBJ, GEJ, JKC, KLC and ILF i.e. - 8 triangles
The bisected triangles are: DGI and JLC i.e. - 2
The triangles divided into three parts are: DGC, DIC, DEK and DKF i.e. - 4 triangles and other triangle: ADC, DBC, DBC, GIC, GHC, HIC i.e. - 5 triangles.
So, the total number of triangles is \[2+8+2+4+5=21\]
9. How many triangles are there in this figure?
(a) 18 (b) 16
(c) 20 (d) 22
Ans. (c)
Explanation: The simplest triangles are ABC, CDE, EFG, GHA, JIM, IML, MLK and JMK i.e. - 8 triangles
The bisected triangles are: JIK, IJL, IKL and JLK i.e. - 4 triangles
Other triangles divided into two parts are: AGM, AMC, GME and MCE i.e. - 4 triangles.
Other triangle: AGC, AGE, ACE and AGC E i.e; - 4 triangles.
So, the total number of triangle is \[8+4+4+4=20\]
10. Count the number of squares in given figure
(a) 12 (b) 15
(c) 16 (d) 18
Ans. (d)
Explanation: The main squares are ABCD, EFGH, IJKL and MNOP i.e. - 4 squares.
The squares divided into two parts are MQRU, QNUS, RUOT and USTP i.e. - 4 squares
The square divide into four parts are: IMUN, MJOU, UOKP and NUPL i.e. - 4 squares
Other squares are: EUU, JUGK, IFUL and ULKH i.e. – 4 squares
So, the total number of squares is: \[4+4+4+4=16\]
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