- #1

Saitama

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- 93

**Problem:**

Let $0<a<b$

i)Show that amongst the triangles with base $a$ and perimeter $a+b$, the maximum area is obtained when the other two sides have equal length $b/2$.

ii)Using the result (i) or otherwise show that amongst the quadrilateral of given perimeter, the square has maximum area.

**Attempt:**

Let $BC=a$ and $AB+AC=b$. From Heron's formula,

$$\Delta =\sqrt{s(s-BC)(s-AB)(s-AC)}$$

Since $s$ is given, we maximise $\sqrt{(s-BC)(s-AB)(s-AC)}$. From AM-GM inequality,

$$\sqrt{(s-BC)(s-AB)(s-AC)}\leq \frac{s^{3/2}}{3\sqrt{3}}$$

For the equality to hold, $s-BC=s-AB=s-AC$ which shows $AB=AC=b/2$. This completes the first part of problem.

How do I start with ii)?

Any help is appreciated. Thanks!