Question

1) K, L, M and N 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.
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Answer 1

Step-by-step explanation:

As ABCD is a square, so

AB = BC = CD = DA     ...(1)

Also

AK = BL = CM = DN   .....(2)

Subtracting (2) from (1)

We get

AB- AK = BC - BL = CD - CM = DA - DN

BK = CL = DM = AN   ...(3)

So now we have

AK = BL = CM = DN

AN = BK= CL = DM

Squaring and adding

AK² + AN² = BL² + BK²= CM²  + CL² = DN² + DM²    ...(4)

But <A = <B = <C = <D = 90°

By Pythagorean theorem (4) Becomes

KN² = KL² = LM² = NM²

So

KN = KL = LM = NM

So KLMN is a rhombus

But <1 = <3 as triangles are congruent

And < 1 + <2 = 90

So <2 + <3 = 90

Hence <KNM = 90°

Therefore KLMN is a square.