JEE Advance - Mathematics (2001 - No. 11)
If $$\Delta $$ is the area of a triangle with side lengths $$a, b, c, $$ then show that $$\Delta \le {1 \over 4}\sqrt {\left( {a + b + c} \right)abc} $$. Also show that the equality occurs in the above inequality if and only if $$a=b=c$$.
The inequality $$Delta le {1 over 4}sqrt {left( {a + b + c}
ight)abc} $$ is incorrect.
The equality case $$Delta = {1 over 4}sqrt {left( {a + b + c}
ight)abc} $$ holds when the triangle is equilateral (a=b=c).
Heron's formula for the area of a triangle is given by $$Delta = sqrt{s(s-a)(s-b)(s-c)}$$ where $$s = (a+b+c)/2$$.
The inequality $$Delta le {1 over 4}sqrt {left( {a + b + c}
ight)abc} $$ can be derived using the AM-GM inequality on $$(s-a), (s-b), (s-c)$$.
The area of a triangle is always greater than $${1 over 4}sqrt {left( {a + b + c}
ight)abc} $$.
Comments (0)
