If a rhombus is re-shaped such that one of its diagonal increases by 4%, while other diagonal decreases by 4%. Find the percentage change in the area of rhombus.


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Answer:

0.16% decrease

Step by Step Explanation:

  1. Let's assume the length of the diagonals BD and AC of the rhombus ABCD are p and q respectively.
  2. The area of the rhombus = 
    pq
    2
     
  3. According to the question, one of its diagonal increases by 4%, while other diagonal decreases by 4%.
    The new length of the diagonal BD = p + p ×  
    4
    100
      = p + 0.04p = (1 + 0.04)p
  4. The new length of the diagonal AC = q - q ×  
    4
    100
      = q - 0.04q = (1 - 0.04)q
  5. Now, the area of the rhombus =  
    (1 + 0.04)p × (1 - 0.04)q
    2
     
    =  
    (12 - 0.042)pq
    2
      ...[Since, (a + b)(a - b) = a2 - b2]
    =  
    pq - 0.0016pq
    2
     
  6. Change in area = New area of the rhombus - The area of the rhombus
    =  
    pq - 0.0016pq
    2
      -  
    pq
    2
     
    =  
    pq - 0.0016pq - pq
    2
     
    =  
    -0.0016pq
    2
     
  7. % Change in area =  
    Change in area
    The area of the rhombus
      × 100
    =  
     
    -0.0016pq
    2
     
     
    pq
    2
     
      × 100
    = -0.16%
  8. Thus, the area of the rhombus is decreased by 0.16%.

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