The diameter MS and two chords MN and MR = R of this circle are drawn through the point M of the circle. Find the angles of the quadrilateral MNSR and the degree measures of the arcs MN, MS, SR.RM.
Since the lengths of the chords MN and MR are equal to the diameter OM, the triangles OMN and OMR are equilateral, and all their internal angles are 60, and the angle BAO = OMN + OMR = 60 + 60 = 120.
Triangles SMN and SMR are rectangular, since their corners SNM and SRM are based on the diameter SM of the circle, then the angle MSN = MRS = (180 – 90 – 60) = 30, and the angle RSN = 30 + 30 = 60.
The central angles MON and MOR are 60 and are based on the arcs MN and MR, the degree measure of which will also be 60.
The degree measures of the arcs SN and SR are equal to the difference between the arcs SM – MN and SM – MR, and are equal to (180 – 60) = 120.
Answer: The angles of the quadrilateral MNSR are 60, 90, 120, 90. The arc MN and MR are 60, the arc SN and SR are 120.
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