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Mathematics

A student wrote the Heron's Formula in his notebook as below.

Area of a triangle with sides a, b and c

= (a+b+c2)(a+b2)(b+c2)(a+c2)\sqrt{\Big (\dfrac{a + b + c}{2}\Big)\Big(\dfrac{a + b}{2}\Big)\Big(\dfrac{b + c}{2}\Big)\Big(\dfrac{a + c}{2}\Big)}.

Is it correct?

Mensuration

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Answer

We know that,

For sides with a, b, c and semi-perimeter 's', Heron's Formula for area of triangle is :

Area of triangle = s(s−a)(s−b)(s−c)\sqrt{s(s - a) (s - b) (s - c)} ⇒(a+b+c2)(a+b+c2−a)(a+b+c2−b)(a+b+c2−c)⇒(a+b+c2)(a+b+c−2a2)(a+b+c−2b2)(a+b+c−2c2)⇒(a+b+c2)(b+c−a2)(a+c−b2)(a+b−c2)\Rightarrow \sqrt{\Big (\dfrac{a + b + c}{2}\Big)\Big(\dfrac{a + b + c}{2} - a\Big)\Big(\dfrac{a + b + c}{2} - b\Big)\Big(\dfrac{a + b + c}{2} - c\Big)} \\[1em] \Rightarrow \sqrt{\Big (\dfrac{a + b + c}{2}\Big)\Big(\dfrac{a + b + c - 2a}{2}\Big)\Big(\dfrac{a + b + c - 2b}{2}\Big)\Big(\dfrac{a + b + c - 2c}{2}\Big)} \\[1em] \Rightarrow \sqrt{\Big (\dfrac{a + b + c}{2}\Big)\Big(\dfrac{b + c - a}{2}\Big)\Big(\dfrac{a + c - b}{2}\Big)\Big(\dfrac{a + b - c}{2}\Big)} \\[1em]

But the given formula is :

(a+b+c2)(a+b2)(b+c2)(a+c2)\sqrt{\Big (\dfrac{a + b + c}{2}\Big)\Big(\dfrac{a + b}{2}\Big)\Big(\dfrac{b + c}{2}\Big)\Big(\dfrac{a + c}{2}\Big)}.

Hence, the student wrote the incorrect formula.

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