Question

K,L M and N are points in the sides AB, BC, CD and DA respectively of square ABCD such that AK=BL=CM=DN.Prove that KLMN is a square.​

Answer 1

solution

                                                                                                           

Given : that AK = BL = CM = DN

ABCD is a square

So we get

BK = CL = DM = AN                                                         (1)

Consider △ AKN and △ BLK

Given : that AK = BL

From the figure we know that ∠ A = ∠ B = 90°

Using equation (1)  = AN = BK

By SAS congruence criterion

△ AKN ≅ △ BLK

We get

∠ AKN = ∠ BLK and ∠ ANK = ∠ BKL (c. p. c. t)

We know that

∠ AKN + ∠ ANK = 90°

∠ BLK + ∠ BKL = 90°

By adding both the equations

∠ AKN + ∠ ANK + ∠ BLK + ∠ BKL = 90° + 90°

On further calculation

2 ∠ ANK + 2 ∠ BLK = 180°

Dividing the equation by 2

∠ ANK + ∠ BLK = 90°

So we get

∠ NKL = 90°

In the same way

∠ KLM = ∠ LMN = ∠ MNK = 90°

Therefore, it is proved that KLMN is a square.

                                                                                                           

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