Proof of Hero’s Formula

Question: Proof of Heros formula for the nub socket of trilateral ABC with sides a, b and c and s being the semi-perimeter i.e. s=a+b+c/2, and so(prenominal) Area A = [pic] Proof: Let a trilateral ABC with sides a, b and c whose bea is equal to A = [pic]. Let the trigon be as follows: Here, perimeter is the length of the sides, now as the sides are a, b & c, assume it is also the length of the sides, indeed the perimeter of this trigon is P = a+b+c And semi-perimeter i.e. half of the perimeter is S = a+b+c/2 If we scuff a perpendicular from C to substructure and call it h which divides the base into two parts i.e. x and c-x, then the plat looks as follows: The perpendicular has divided the triangle into two right-angled triangles. Now for any right-angle triangle, according to Pythagorean Theorem, [pic] = [pic] + [pic] If Pythagorean is em ploy to the right-angled triangles in the above triangle, then in the fictitious character of left over(p) right-angle triangle in the above diagram, it would fertilise us the equality [pic] = [pic] + [pic] where a = hypotenuse and h = height/perpendicular and x = base. Re-writing it, the equation would become which we ordain call Eq.

A [pic] = [pic] - [pic] ---------------------( Eq. A Similarly, for the right angle triangle on the right half to triangle ABC, [pic] = [pic] + [pic] where b = hypotenuse, h = height/perpendicular and c-x = base. Re-writing this equation would result in [pi c] = [pic] [pic] Expanding [pic] would! give us [pic] = [pic] ([pic] + [pic] - 2cx) [pic] = [pic] [pic] - [pic] + 2cx --------------------------( Eq. B As, the left hand sides of Eq.A and Eq. B are equal, we can equate them. equate Eq.A and Eq. B would give us [pic] - [pic] = [pic] [pic] - [pic] + 2cx Solving it further would give us [pic] = [pic] [pic] + 2cx Re-arranging...If you emergency to drag a full essay, order it on our website: BestEssayCheap.com

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