Question

DERIVATION OF HERONS FORMULA??? PLZZ ANYONE EXPLAIN ME ..STEP BY STEP

Answer 1

See diagram.  Apply Pythagoras theorem.

let    s = (a+b+c)/2

c² = h² + x²   ---(1)
b² = h² + (a-x)²
    = h² + a² - 2ax + x² 
    = h² + a² - 2 a x + c² - h²
x = (a² + c² - b²) / 2a
 
Substitute value of x in (1)   h² = c² - (a² + c² - b²)²/4a² 
h²  = [ (2ac)² - (a² + c² - b²)² ] / 4a²
    =  [ (2ac +a² + c² - b²) (2ac - a² - c² + b²) ] /4a²
    =   [ (a + c + b)(a+c - b) ( b - a + c) ( b + a - c) ] / 4 a² 
    =  [ (2 s) (2s - 2b) (2s - 2a) (2s - 2c)]/4a²

Area of ΔABC = 1/2 * a * h = √[s(s-a)(s-b)(s-c)

proved.