ABCDEF is a regular hexagon inscribed in a circle with centre O. Show that each side of it subtends an angle of 60 degrees at its centre. Also show that each of its sides is equal to the radius of its circle.
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Since, ABCDEF is a regular hexagon, all the sides are equal. The lines joining the center to the vertices OA, OB, OC, .. OF divide the angle at O in equal proportions. Hence angle 360 deg. is divided in to 6 equal parts.
Hence AOB, AOF, BOC, COD, DOE, EOF each = 360/6 = 60 deg
In the triangle OAB, OA = OB, as they are equal to radius of circle. Hence OAB is an isosceles triangle. Hence OBA = OAB .
OAB = (180 - AOB )/2 = 60 deg
Hence all angles in OAB are equal. Hence it is an equilatera l triangle. Hence AB = radius of circle.
In other triangles also, you can prove in a similar way, OBC= OCB and BC = radius. and so on. Hence, all sides are equal to the radius of the circle.
Hence AOB, AOF, BOC, COD, DOE, EOF each = 360/6 = 60 deg
In the triangle OAB, OA = OB, as they are equal to radius of circle. Hence OAB is an isosceles triangle. Hence OBA = OAB .
OAB = (180 - AOB )/2 = 60 deg
Hence all angles in OAB are equal. Hence it is an equilatera l triangle. Hence AB = radius of circle.
In other triangles also, you can prove in a similar way, OBC= OCB and BC = radius. and so on. Hence, all sides are equal to the radius of the circle.
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