counting figures model question set 3 Detailed Explanation And More Example

Answer: (b)

From the figure, it is clear that there are 9 triangles in the upper half of the figure ABFE, namely ΔPEM, ΔPGI, ΔAEI, ΔEIN, ΔIKQ, ΔQOF, ΔIBF, ΔINF and ΔIEF.

Likewise, there are 9 triangles in the lower half EFCD of the main figure.

There are more triangles ΔEPR, ΔEIJ, ΔIFJ and ΔQFS.

Hence, there are a total of 22 triangles in the figure.

Answer: (a)

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.

∴ Total number of triangles in the given figure.

9 + 9 + 1 + 4 +2 + 1 + 1 = 27.

Answer: (b)

9 Quadrilaterals are as follow

□AGDH, □BAFD, □CAED, □AGDF, □AGDE, □ABDH, □ACDH, □ACDF and □ABDE

Answer: (a)

The figure, in question has been labelled at different points as shown in the following figure

From the above diagram parallelograms are as follow

i) Parallelograms from the outer figure = ▱ABCD, ▱AIJD, ▱GBCH, ▱ADHG, ▱GHJI, ▱IJCB = 6

ii) Parallelograms from the upper half of the figure = ▱AGKE, ▱GILK, ▱IBFL, ▱AILE, ▱GBFK, ▱ABFE = 6

iii) Parallelograms from the lower half of the figure = ▱EKHD, ▱KLJH, ▱LFCJ, ▱KFCH, ▱ELJD, ▱EFCD = 6

∴ there are a total of 18 parallelograms in the figure.

Shortcut Method

Number of parallelograms = (3 + 2 + 1) × (2 + 1)

[∵ C = 3, R = 2]

= 6 × 3 = 18

Answer: (a)

There are a total of 19 squares in the figure, which have been marked as shown in the following figure.

There are three suqares in the upper row □EFIH, □FGJI, □GBLJ.

There are five suqares in the middle row namely □TQRS, □QHPR, □HIOP, □IJNO, □JLMN

There are three suqares in the lower row namely □SRUD, □RPVU, □POWV

By combining two upper rows there are three suqares namely □AEPS, □EGNP, □FBMO.

By combining two lower rows there are three suqares namely □ THVD, □ QIWU, □ ILCW

From all the three rows combined together there are two suqres namely □ AFWD and □ EBCV

∴Total number of squares = 3 + 5 + 3 + 3 + 3 + 2 = 19

Answer: (c)

We may the label the figure 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.

Thus, there are 6 + 4 = 10. triangles in the given figure.