Consider a square ABCD of area 25 cm2. L is the midpoint of AB, M the midpoint of BC, N the midpoint of CD, and O the midpoint of DA. These points are used to construct a new square LMNO. The same process is repeated on LMNO to construct a smaller square QRST (where Q is the midpoint of LM and so on). What is the perimeter of square QRST?


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Answer:

10

Step by Step Explanation:
  1. According to the question, area of the square ABCD = 25 cm2
    Given, L is the midpoint of AB, M the midpoint of BC, N the midpoint of CD, and O the midpoint of DA. These points are used to construct a new square LMNO. The same process is repeated on LMNO to construct a smaller square QRST (where, Q is the midpoint of LM and so on).
    The following figure shows the mentioned constructions:
  2. Let us assume a as the side of the square ABCD. Since, the square ABCD has the area 25 cm2.Therefore, we can say that a2 = 25
    ⇒ a = √25 cm2
    ⇒ AB = a = √25 cm
    Since, L and O are the midpoints of AB and AD, respectively, therefore AL = AO =  
    √25
    2
      cm
  3. Now, in the right angle triangle ΔALO
    OL2 = AL2 + AO2
    ⇒ OL2 = ( 
    √25
    2
     )2 + ( 
    √25
    2
     )2
    ⇒ OL2 =  
    25
    4
      +  
    25
    4
     
    ⇒ OL2 =  
    50
    4
     
    ⇒ OL =  
    √50
    √4
     
    ⇒ OL =  
    √50
    2
      cm
    Now, the side of square LMNO is  
    √50
    2
      cm
    Since, Q and T are the midpoints of LM and LO respectively.
    Therefore, LT = LQ =  
    √50
    4
      cm
  4. Similarly, in the right angle triangle ΔLQT,
    QT2 = LT2 + LQ2
    ⇒ QT2 = ( 
    √50
    4
     )2 + ( 
    √50
    4
     )2
    ⇒ QT2 = ( 
    50
    16
     ) + ( 
    50
    16
     )
    ⇒ QT2 = ( 
    100
    16
     )
    ⇒ QT2 = ( 
    25
    4
     )
    ⇒ QT =  
    5
    2
      cm
  5. Thus, the perimeter of the square QRST = 4 × QT
    = 4 ×  
    5
    2
     
    = 10 cm

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