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Worked Examples

Example-1
How many triangles are there in the adjacent figure?
 
Description: 61918.png
  1. 15
  2. 19
  3. 17
  4. 20
Solution
First, label the figure as shown below and then count the triangles.
 
There are 19 triangles.
 
Description: 61951.png
 
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?
 
Description: 62025.png
  1. 25
  2. 28
  3. 29
  4. 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?
 
Description: 62064.png
  1. 25
  2. 27
  3. 24
  4. 26
Solution
First, label the diagram as A, B, C, D, … 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.
 
Answer: (4)

Alternatively, by applying the formula we get,
 
Total number of squares = m.n + (m - 1) (n - 1) + . . .
 
= (5 × 3) + (4 × 2) + (3 × 1) = 15 + 8 + 3 = 26.
 
 
Example-4
How many parallelograms are there in the figure?
 
Description: 62130.png
  1. 16
  2. 29
  3. 30
  4. 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.
 
Answer: (3)

We can also use the formula and find out the total number of parallelograms.
 
Total number of parallelograms = Description: 62117.png.
 
 
Example-5
How many squares are there in the following figure?
 
Description: 62156.png
  1. 13
  2. 17
  3. 23
  4. 27
Solution
Each side of the bigger square is divided into three equal parts.
Apply the formula: If a square is divided into ‘n’ parts on each side, then the total number of squares formed = Description: 62177.png
Here the value of n = 3.
So, the total number of squares = Description: 62193.png 
(Or to be simple, add 12 + 22 + 32 = 1 + 4 + 9 = 14.)
Squares at the four corners = 4 × 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.
 
Description: 62223.png
  1. 23
  2. 21
  3. 25
  4. 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.
Description: 62241.png
  1. 25
  2. 18
  3. 21
  4. 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)
 




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