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Prove that the isosceles triangle whose range is the largest of the isosceles triangles that can be inscribed within a given circle |
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Answer» Step-by-step explanation: Explanation: Let their be an ISOSCELES triangle ABC inscribed in a circle as SHOWN, in which equal sides A C and B C subtend an angle x at the center. It is apparent that side A B subtends an angle 360 0 − x at the center (as shown). NOTE that for equilateral triangles all these angles will be 2 π 3 . As the area of the triangle portion subtended by an angle x is R 2 2 sin x , the complete area of triangle ABC is A = R 2 2 ( sin x + sin x + sin ( 360 − 2 x )
= R 2 2 ( 2 sin x − sin 2 x )
= R 2 ( sin x − sin x cos x )
= R 2 sin x ( − cos x )
For maximization we should have d A d x = 0
i.e. R 2 ( cos x ( 1 − cos x ) + sin x × sin x ) = 0
or cos x − cos 2 x + 1 − cos 2 x = 0
or 2 cos 2 x − cos x − 1 = 0
or 2 cos 2 x − 2 cos x + cos x − 1 = 0
or 2 cos x ( cos x − 1 ) + 1 ( cos x − 1 ) = 0
or ( 2 cos x + 1 ) ( cos x − 1 ) = 0
cos x = − 1 2 or cos x = 1
i.e. x = 2 π 3 or x = 0
But for a triangle x ≠ 0
hence x = 2 π 3
and hence for maximum area triangle must be equilateral. |
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