In the figure, if LK = 6√3 , MK = 12, find the remaining sides of □LMNK and also perimeter of □ LMNK.
Solution:
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In ∆ MLK, m∠ L = 900 , m∠ K = 300
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∴ m∠ M = 600 [Remaining angle
of a triangle]
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∴ ∆ MLK, is a 300 – 600
– 900 triangle
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Now, Side opposite to
300 = 1/2
[Hyp]
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∴ ML = ½ [MK]
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∴ ML = ½ [12]
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∴ ML = 6 cm
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Now, in ∆ MNK, ∠ N = 450 , ∠ K = 900 [Given]
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∴ ∠ M = 450 [Remaining angle
of a triangle]
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∴ MNK, is a 450 , 45‑0,
900 triangle
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∴ side opposite to 450 = 1/√2 [Hyp]
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∴ MK = KN = 1/√2 [MN]
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∴ 12 = KN = 1/√2 [MN]
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∴ KN = 12
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&
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12 = 1/√2 [MN]
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&
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12√2 = MN
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P(□ LMNK) = LM + MN + NK + LK
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∴ P(□ LMNK) = 6 + 12√2 +12 + 6√3
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∴ P(□ LMNK) = 18 + 12√2 + 6√3
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∴ P(□ LMNK) = 6(3 + 2√2 +√3) cm
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