Q. K, L, M and N …
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Gaurav Seth 5 years, 1 month 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
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.
1Thank You