In a rectangle ABCD, E, F, G, and H are the midpoints of the four sides, what kind of shape is represented by EFGH.


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Answer:

Rhombus

Step by Step Explanation:
  1. The following figure shows the rectangle ABCD,

    Let's assume the length and breadth of the rectangle ABCD be a and b respectively.
  2. In ΔGDH, DG =  
    CD
    2
      =  
    a
    2
     ,
    DH =  
    DA
    2
      =  
    b
    2
       [Since, G and H are the midpoints of the sides CD and DA respectively.]
    ∠D = 90°  [Since, ABCD is a rectangle.]
    GH2 = DG2 + DH2  [By the pythagorean theorem.]
    ⇒ GH2 = ( 
    a
    2
     )2 + ( 
    b
    2
     )2
    Similarly, HE2 = EF2 = FG2 = ( 
    a
    2
     )2 + ( 
    b
    2
     )2
    and hence, HE2 = EF2 = FG2 = GH2,
    or HE = EF = FG = GH
  3. We know that quadrilateral with all four sides equal is rhombus. Therefore, EFGH is a Rhombus.

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