## Monday, May 28, 2012

### IX Proof of Heron’s formula

Let a, b, c are length of the sides and h is height to side of length c  of ∆  ABC.
We have S = (a + b + c)/2
So, 2s = a + b + c
Þ  2(s - a) = - a + b + c
Þ  2(s - b) = a - b + c
Þ  2(s - c) = a + b – c
Let p + q = c as indicated.
Then,   h2 =  a2 - p2       -------------(1)
Also,   h2 = b2 - q 2       -------------- (ii)
From (i) and (ii)
Þ  a2 - p2     =   b2 -  q 2
Þ  q2   = - a2 + p2 + b2
Since,   q = c - p  Þ q2 = (c-p)2 Þ q2  = c2 + p2 -2pc
Then, c2 + p2 -2pc   = - a2 + p2 + b2
Þ - 2pc  =  - a2 +b2 – c2 = - ( a2 -b2 + c2)
Þ   p = ( a2 -b2 + c2)/2c
Now, Put this value of p in equation (i)
h2    =   a2 -  p2
h2      =   ( a – p ) ( a + p )
h2   =    {a – ( a2 -b2 + c2)/2c }   {a + ( a2 -b2 + c2)/2c }
h2   =   {(2ac - a2 + b2 - c2)/2c}x{(2ac+ a2 - b2 + c2)/2c}
h2      = {(b2 – (a - c)2 }{(a + c)2 – b2}/4c2
h2     = {(b – a + c) (b + a - c)}{(a + c + b)(a + c – b )
h2 = { 2(s - a) x 2(s - c)  x 2s x2(s - b)}/4c2
h2 = { 4 s (s - a) x (s - c) x(s - b)}/c2
h = 2/c   s (s - a) x (s - b) x(s - c)
½ h c =  s (s - a) x (s - b) x(s - c)
Area of triangle =  s (s - a) x (s - b) x(s - c)

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1. CBSE Area of triangles by Heron's formula

9th Area of triangles by Heron's formula Test Paper-1