The figure formed by joining the mid-points of the adjacent sides of a square is a ____.

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The figure formed by joining the mid-points of the adjacent sides of a square is a square.Construct a square

ABCD
 . Join the midpoints

E
  (of

AD
 ),

F
  (of

AB
 ),

G
  (of

BC
 ), and

H
  (of

DC
 ) as shown in the figure given below.Consider

ΔAEF
 ,

ΔGHC
 ,

ΔDEH
 , and

ΔBFG
 . All sides of a square

AD,BC,AB and CD
  are equal in length, and all the angles

∠FAE,∠FBG,∠GCH
 ,

∠EDH
  are right angles. Therefore, sides:

AD,BC,AB and CD
  are equal.

1
    2
   
   AD,
    1
    2
   
   BC,
    1
    2
   
   AB and 
    1
    2
   
   CD are equal. 
 And angles

∠FAE,∠FBG,∠GCH
  and

∠EDH
  are equal. Thus, by SAS congruency,

ΔAEF,ΔGHC and ΔDEH,ΔBFG
 are congruent. Therefore, sides

EF,GH,EG and FG
  are equal.                                  ……(1)Consider

ΔEDH
 ,

ED=DH
  and it is an isosceles triangle.

∠DEH=∠DHE=x
 By angle sum property of the triangle,

∠DEH+∠DHE+HDE=180°

⇒2x+90°=180°

⇒x=
    
     45
    °

......(2) Consider

ΔAEF
 ,and it is an isosceles triangle.

∠AEF=∠AFE=y
 By angle sum property of the triangle,

∠AEF+AFE+FAE=180°

⇒2y+90°=180°

⇒y=
    
     45
    °

.......(3) Analyze from equations (2) and (3) and substitute the values of

x,y
 .By angle sum property of the triangle,

x+y+∠FEH

=
        
         180
        °

90
        °
       
       +∠FEH

=
        
         180
        °

∠FEH

=
        
         90
        °

Similarly,

∠EHG,∠FGH,∠EFG
  are all

90
    °

. Thus,

EFGH
  is a square.Hence, the required answer is square.

The figure formed by joining the mid-points of the adjacent sides of a square is a ____.

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