# Worked Examples

Example-1

How many triangles are there in the adjacent figure?

- 15
- 19
- 17
- 20

Solution

First, label the figure as shown below and then count the triangles.
There are 19 triangles.

AEH | AEB | EBF | ABH | BHI | FGI | GJC | GIJ | BIC | AJC |

CIJ | CJD | JDK | AJH | AJD | EHJ | FIJ | AFB | AGC |

**Answer:**(2)Example-2

How many rectangles (excluding squares) are there in the figure?

- 25
- 28
- 29
- 30

Solution

There are 29 rectangles in the above figure. The rectangles are:

ABFE | BCGF | ACGE | EFIH | FGJI | EGJH | HIML | IJNM | HJNL | ACNL |

LMPO | MNQP | LNQO | ACJH | EGNL | HJQO | BDKI | IKRP | ADKH | EGQO |

HKRO | ABML | EFPO | ABPO | BCNM | FGQP | BCQP | CDRQ | ADRO |

**Answer:**(3)Example-3

How many squares are there in the adjacent figure?

- 25
- 27
- 24
- 26

Solution

First, label the diagram as A, B, C, D, â€¦ X as shown in the figure.
There are five squares in the top strip of the figure, namely ABHG, BCIH, CDJI, DEKJ and EFLK.
Similarly, there are five squares in each of the middle and bottom strips.

So, there are 15 squares if a single strip is considered at a time.
Now, consider two strips at a time. There are eight squares (ACOM, BDPN, CEQO, DFRP in the strip AFRM and GIUS, HJVT, IKWU, JLXV in the strip GLXS).
Finally, consider all the three strips together. There are three squares ADVS, BEWT and CFXU.
Thus, the total number of squares = 15 + 8 + 3 = 26.

Alternatively, by applying the formula we get,
Total number of squares = m.n + (m - 1) (n - 1) + . . .
= (5 Ã— 3) + (4 Ã— 2) + (3 Ã— 1) = 15 + 8 + 3 = 26.

So, there are 15 squares if a single strip is considered at a time.

**Answer:**(4)Alternatively, by applying the formula we get,

Example-4

How many parallelograms are there in the figure?

- 16
- 29
- 30
- 26

Solution

First, label the different points in the figure as shown.
Parallelograms in the outer figure are ABLK, BCML, CDNM, DEON, ACMK, BDNL, CEOM, ADNK, BEOL and AEOK.
In the top horizontal strip, there are 10 parallelograms. They are:

ABGF, BCHG, CDIH, DEJI, ACHF, BDIG, CEJH, ADIF, BEJG and AEJF.
Similarly, in the bottom horizontal strip, the following 10 parallelograms are there.
FGLK, GHML, HINM, IJON, FHMK, GINL, HJOM, FINK, GJOL and FJOK.
So, total number of parallelograms = 10 + 10 + 10 = 30.

We can also use the formula and find out the total number of parallelograms.
Total number of parallelograms = .

ABGF, BCHG, CDIH, DEJI, ACHF, BDIG, CEJH, ADIF, BEJG and AEJF.

**Answer:**(3)We can also use the formula and find out the total number of parallelograms.

Example-5

How many squares are there in the following figure?

- 13
- 17
- 23
- 27

Solution

Each side of the bigger square is divided into three equal parts.

Apply the formula: If a square is divided into â€˜nâ€™ parts on each side, then the total number of squares formed =

Here the value of n = 3.

So, the total number of squares =

(Or to be simple, add 1

Squares at the four corners = 4 Ã— 2 = 8.

Square formed at the centre = 1.

Total number of squares = 14 + 8 + 1 = 23.

Apply the formula: If a square is divided into â€˜nâ€™ parts on each side, then the total number of squares formed =

Here the value of n = 3.

So, the total number of squares =

(Or to be simple, add 1

^{2}+ 2^{2}+ 3^{2}= 1 + 4 + 9 = 14.)Squares at the four corners = 4 Ã— 2 = 8.

Square formed at the centre = 1.

Total number of squares = 14 + 8 + 1 = 23.

**Answer:**(3)Example-6

Find the number of triangles in the figure.

- 23
- 21
- 25
- 29

Solution

IJK | JKL | JIL | CIJ | CDJ | DEJ | CEJ | EJL | FGH | BCG |

AFC | AEM | FMI | CEL | CEI | EIL | CFI | CFL | CIL | MNO |

MIP | MJQ | MER | HIN | BEN |

Total number of triangles = 25.

**Answer:**(3)

Example-7

Find the total number of triangles in the figure.

- 25
- 18
- 21
- 20

Solution

There are 18 triangles as furnished below:

ADE | ABE | BEF | BFC | CFG | DHI | DEI | EFI | FGI |

IGJ | ADI | AHI | CIG | CIJ | ACI | DGI | EIG | DFI |

**Answer:**(5)