Question

L, M, N, K are mid-points of sides BC, CD, DA and AB respectively of square ABCD, Prove that LMNK form a rhombus.​

Answer 1

Answer

Proved Below

Step-by-step explanation:

Given:

Here, L, M, N, K are mid-points of sides BC, CD, DA and AB respectively of square ABCD.

Construction:

Join BN, DL, KD, BM.

Proof:

As shown in the figure below, L, M, N, K are mid-points,

⇒ BN║DL and BQ║DK

Therefore DNBQ is a parallelogram.

⇒Δ ABN ≅ Δ ADK   [SAS = SAS congruency]        

⇒ ∠ ABN = ∠ ADK   [By CPCT]                    [1]

Also, Δ PND ≅ Δ PKB           [ASA congruency]

Therefore PB = PD    [By CPCT]                   [2]

So DQBP is a rhombus        [ from 1 and 2 ]

Hence DL, DK, BM and BN enclose a rhombus.

Hence proved.