A) 64 | B) 32 |

C) 24 | D) 16 |

Explanation:

When the triangles are drawn in an octagon with vertices same as those of the octagon and having one side common to that of the octagon, the figure will appear as shown in (Fig. 1).

Now, we shall first consider the triangles having only one side AB common with octagon ABCDEFGH and having vertices common with the octagon (See Fig. 2).Such triangles are ABD, ABE, ABF and ABG i.e. 4 in number.

Similarly, the triangles having only one side BC common with the octagon and also having vertices common with the octagon are BCE, BCF, BCG and BCH (as shown in Fig. 3). i.e. There are 4 such triangles.

This way, we have 4 triangles for each side of the octagon. Thus, there are 8 x 4 = 32 such triangles.

A) 5 | B) 2 |

C) 3 | D) 6 |

Explanation:

The given figure can 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 colour pencils are required.

A) 20 | B) 19 |

C) 17 | D) 15 |

Explanation:

The given figure can be labelled as :

**Straight lines** :

The number of straight lines are 19

i.e. BC, CD, BD, AF, FE, AE, AB, GH, IJ, KL, DE, AG, BH, HI, GJ, IL, JK, KE and DL.

A) 18 | B) 17 |

C) 14 | D) 16 |

Explanation:

The given figure can 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.

A) 5 | B) 8 |

C) 9 | D) 10 |

Explanation:

The given figure can be labelled as :

The simplest triangles are AJF, FBG, HDI, GCH and JEI i.e 5 in number.

The triangles composed of the three components each are AIC, FCE, ADG, EBH and DJB i.e 5 in number.

Thus, there are 5 + 5 = 10 triangles in the given figure.

A) 21 | B) 15 |

C) 18 | D) 20 |

Explanation:

The simplest triangles are AKI, AIL, EKD, LFB, DJC, DKJ, KIJ, ILJ, JLB, BJC, DHC and BCG i.e. 12 in number.

The triangles composed of two components each are AKJ, ALJ, AKL, ADJ, AJB and DBC i.e. 6 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 12 + 6 + 2 + 1 = 21 triangles in the figure.

A) 30, 5 | B) 32, 3 |

C) 28, 5 | D) 30, 3 |

Explanation:

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 CVWM i.e 4 in number.

The rectangles composed of three components each are ACRP, PRMO, EGHT and THIK i.e 4 in number.

The rectangles composed of four components each are CEKM, AVSP, PSWO,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.

A) 36 triangles, 7 Squares | B) 38 triangles, 9 Squares |

C) 40 triangles, 7 Squares | D) 42 triangles, 9 Squares |

Explanation:

The figure may be labelled as shown

**Triangles :**

The Simplest triangles are BGM, GHM, HAM, ABM, GIN, IJN, 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, IKJL and KDEL i.e 4 in number

Total number of squares in the figure = 3 + 4 =7