# Understand the problem

Show that among all quadrilaterals of a given perimeter the square has the largest area.

##### Source of the problem

Indian National Mathematical Olympiad 1986

##### Topic

Geometry

##### Difficulty Level

Easy

##### Suggested Book

An Excursion in Mathematics

# Start with hints

Do you really need a hint? Try it first!

Start with a quadrilateral with sides and . Divide it into two triangles and write its area as the sum of the areas of the triangles.

Show that the area satisfies and .

Using hint 2, derive that .

From the AM-GM inequality, we can write that . Hence . Equality is achieved iff all the angles are right angles (this follows from hint 1) and . If all the angles are right angles then the quadrilateral is a rectangle and hence and . Finally, . Thus the area is maximised for a square.

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# Connected Program at Cheenta

#### Math Olympiad Program

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.