Show that the least perimeter of an isosceles triangle
Answers
Answer:
ABC is an isosceles triangle with sides AB=AC. O is the center of circle of radius inscribed in the triangle. AD is the altitude,
So OD=OF=OE=r
So BD = CD ............. (1) ( altitude bisect the side)
BD = BE and CD = CF .............. (2) ( tangents from the same point are equal in length)
From (1) and (2)
BD=BE=CD=CF ........ (3)
Similariy AE = AF ........... (4) (tangents from the same point)
Perimeter of triangle (P) = AB + BC + AC
(P) = AE + EB + BD + CD + CF+AF
(P) = AE + AF + EB + BD + CD + CF
(P) = 2AE + 4BD (using 3 and 4)
In Right Triangle OEA,
AE = OETan x
=> AE = rtan x
=> AE = r cot x .............. (5)
In Right Triangle ABD
BD = AD*tanx
=> BD = (AO + OD)* tan x
=>BD = rsin x + rtan x
=> BD = r secx + r tanx ............... (6)
From 5 and 6
=> (P) = 2r cotx + 4r(sec x + tanx)
=> (P) = r ( 2 cotx + 4 sec x+ 4 tanx)
Differentiate w.r.t. x
we get
dPdx = r( -2cosec2x + 4 sec x tan x + 4 sec2x) ........ (7)
=> dPdx = r( -2sin2x + 4sin xcos2x + 4cos2x)
To find critical value, Put dPdx = 0
=> r( -2sin2x + 4sin xcos2x + 4cos2x) = 0
=> ( -2sin2x + 4sin xcos2x + 4cos2x) = 0
=> -2cos2x + 4sin3x + 4sin2x sin2x ×cos2x = 0
=> -2(1-sin2x)+ 4sin3x + 4sin2x = 0
=> -2+2sin2x+ 4sin3x + 4sin2x = 0
=> -2+ 4sin3x + 6sin2x = 0
=>2sin3x + 3sin2x - 1 = 0
=>(2sin3x + 2sin2x) + (sin2x - 1) = 0
=>2sin2x(sin x + 1) + (sinx +1) (sin x - 1) = 0
=> (sin x + 1) (2sin2x + sin x - 1) = 0
=> (sin x + 1) = 0 or (2sin2x + sin x - 1) = 0
=> sin x = -1 is not possible because x cant be more than 90° so (2sin2x + sin x - 1) = 0
=> 2sin2x + 2sin x - sin x - 1 = 0
=> 2sinx( sin x + 1) - 1( sin x + 1) = 0
=> (2sinx- 1) ( sin x + 1) = 0
=> (2sinx- 1) = 0 or ( sin x + 1) = 0
=> sin x = -1 is not possible because x cant be more than 90° so (2sin x - 1) = 0
=> 2sinx- 1= 0
=> sinx = 12
=> x = 30°
at sinx = 12 sign changes from -ve to +ve. so it is point of local minima, so at this point value ll be least.
Hence least perimeter is
(P) = r ( 2 cot 30° + 4 sec 30°+ 4 tan30°)
=> (P) = r ( 2 3 + 4×23+ 4 13)
=> (P) = r 6+ 8+43
=> (P) = r 183
=> (P) = 63r
Hence Proved
hopw it will help you...