We have S = (a + b + c)/2

So, 2s = a + b + c

2(s - a) = - a + b + c

2(s - b) = a - b + c

2(s - c) = a + b – c

h2 = a2 - p2 -------------(1)

Also, h2 = b2 - q 2 -------------- (ii)

From (i) and (ii) Þ a2 - p2 = b2 - q 2

q2 = - a2 + p2 + b2 Since, q = c - p

q2 = (c-p)2 Þ q2 = c2 + p2 -2pc

Then, c2 + p2 -2pc = - a2 + p2 + b2

- 2pc = - a2 +b2 – c2 = - ( a2 -b2 + c2)

p = ( a2 -b2 + c2)/2cNow, 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}/4c

2h2 = {(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)

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

9th Area of triangles by Heron's formula Test Paper-2

9th Area of triangles by Heron's formula Test Paper-3

Assignment for Class IX On : HERONS FORMULA

9th Heron formula proof and test questions

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