In the convex quadrangle NPQM, the diagonal NQ is the bisector of the angle PNM and intersects the diagonal PM at the point S. Find NS if it is known that a circle can be described near the quadrangle NPQM, PQ = 44, SQ = 22
∠QNM – is inscribed in a circle and relies on the arc QM.
∠QPM is also inscribed in a circle and rests on the arc QM.
Therefore, these angles are equal.
∠QNM = ∠QPM
Consider the triangles NPQ and SPQ.
∠SQP – General
∠QNP = ∠SPQ
According to the first sign of the similarity of triangles, these triangles are similar.
Then, NQ / QP = QP / SQ
NQ = QP ^ 2 / SQ = 44 ^ 2/22 = 88
NS = NQ-SQ = 88-22 = 66
Answer: NS = 66
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