Largest possible area of the triangle ABC such that AB≤2, BC≤3, AC≤4.

Question

Find the largest possible area of the triangle $ABC$ such that $AB$ ≤ 2, $BC$ ≤ 3, and AC ≤ 4.

Note: The solution should not contain any calculus method

Can someone tell me how to maximize the area of a triangle? Do we have to make the sides as close as possible to each other to get the maximum area?

Answer 1

Since $AB$ and $BC$ have lower upper bounds, they should be set to maximum. Then if $AC$ is also set to maximum, angle $B$ becomes blunt ($2^2+3^2<4^2$), which is not good. So we set angle $B$ to $90^\circ$ and the maximum area should be $\frac{1}{2}\times 2\times 3=3$. Once the maximum is found (guessed), it's not hard to prove it:

$$S=\frac{1}{2}|AB||BC|\sin B\leq\frac{1}{2}|AB||BC|\leq\frac{1}{2}\times 2\times 3=3.$$

No calculus. Just a simple trigonometry.

Answer 2

you can also use Heron's formula. Heron's formula states that the area of a triangle whose sides have lengths a, b, and c is :- Put the upper limits of a=2,b=3,c=4