Question

K,L,M andN are points on the sides AB, BC, CD and DA
respectively of a square ABCD such that Ak= BL= CM=DN. prove that KLMN is a square

Answer 1

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Answer 2

Quadrilateral KLMN is a square hence proved.

• It is given that ABCD is square.
• Points K,L,M and N are on sides AB,BC,CD and AD respectively such that AK=BL=CM=DN.
• As ABCD is a square AB=BC=CD=AD.

AK+KB=BL+LC=CM+MD=DN+NA

• Now , AK=BL=CM=DN therefore KB=LC=MD=NA

• That means K,L,M and N are bisectors of AB,BC,CD and AD.

• Now we consider ΔNAK and ΔLBK

NA= BL  (Equal sides)

AK = BK   (Equal sides)

∠A = ∠B (Each angle of a square is 90°)

• Therefore, ΔNAK and ΔLBK are congruent. (SAS test of congruence)

• Now, as ΔNAK ≅ ΔLBK

NK=LK    

• Similarly for other triangles ΔMCL, ΔMDN ,ΔNAK and ΔLBK are all congruent.

• Therefore,

NK=LK=LM=MN   (Using congruent triangles).

• Now the angle formed at K= 45°+∠NKL+45°=180 (Other angle is 45° as side opposite to equal sides in a right angled triangle is 45°)

∠NKL = 90°

Similarly for other angles ,

∠NKL = 90°,∠KNM = 90° ,∠LMN = 90° and ∠LKM = 90°

• Therefore quadrilateral KLMN is a square as all sides are equal and each angle of the quadrilateral is 90°.