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Q. K, L, M and N …

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

Posted by Ankit Raj 6 years, 3 months ago

CBSE > Class 09 > Mathematics

1 answers

Gaurav Seth 6 years, 3 months ago

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

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.

1Thank You

ANSWER

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