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Segment LM in a 30-60-90 triangle

High school · ●●●○○ 50 pts

In triangle ABCABC, ∠B=30∘\angle B=30^{\circ} and ∠A=90∘\angle A=90^{\circ}. On side ACAC, point KK is marked, and on side BCBC, points LL and MM are marked such that KL=KMKL=KM (point LL lies on segment BMBM). Given AK=4AK=4, BL=31BL=31, MC=3MC=3, find the length of segment LMLM.

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isosceles trianglesright triangle trigonometrysegment lengths