# In the convex quadrangle NPQM, the diagonal NQ is the bisector of the angle PNM and intersects the diagonal PM

May 31, 2020 | Education

| **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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