Prove that the isosceles triangle whose range is the largest of the isosceles triangles that can be inscribed within a given circle
Answers
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
(
1
−
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
Hence
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.
