|
|
|
|
|
| Author |
Message |
![[Post New]](/templates/default/images/icon_minipost_new.gif) 25 Jan 2007 14:15:15 IST
|
|
|
SHOW THAT THE TRIANGLE OF MAX. AREA THAT CAN BE INSCRIBED IN A GIVEN CIRCLE IS AN EQUILATERAL TRIANGLE ???
|
|
|
|
25 Jan 2007 15:32:23 IST
|
|
|
Let ABC be a triangle inscribed in the circle with center O and radius r.
If the area of this triangle is maximum, then vertex C should be at a maximum distance from the base AB i.e. CD must be perpendicular to AB.
Hence, ABC is an isosceles triangle.
If  BCD =  , where D is the mid-point of BC, then BOD = 2
so, AB = 2 BD
= 2r sin 2
CD = CO + OD = r + r cos 2
If S be the area of the triangle ABC, then
S = (1/2) AB x CD
= (1/2) x 2r sin 2 (r + r cos 2)
ds/d = r2[sin2 (-2 sin2) + (1 + cos2)(2 cos2)]
= 2r2[cos22 - sin22 + cos2] = 2r2(cos4 + cos2)
For maximum and mimimum
ds/d = 0
or, cos4 + cos2 = 0
or, 2 cos3 cos = 0
so, Either cos3 = 0 or, cos =0
If cos = 0 or 3 =  /2 or = /6
(d2s/d2) = -ve, for = /6
ACB = 2 = 2(/6) = /3 = ABC = BAC
so  ABC is an equilateral triangle.
|
The Scientist does not study nature because it is useful; he studies it because he delights in it, & he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, life would not be worth living. Ofcourse I do not here speak of that beauty that strikes the senses, the beauty of qualities & appearances; not that I undervalue such beauty, far from it, but it has nothing to do with science; I mean that profounder beauty which comes from the harmoniuos order of the parts, & which a pure intelligence can grasp. |
this reply: 7 points
(with 1 
in 2 votes ) [?]
|
|
You have to be logged on to rate
|
|
|
|
|
|
|
|
|
|
Moderation Team
![[Up]](/templates/default/images/icon_up.gif "[Up]"){kind=icon}
