Maximum Area

Geometry Level 5

Given that the three sides of a triangle have lengths a , b , c a,b,c units respectively, where a , b , c a,b,c are integers. If the perimeter is P P units, the area is A A square units, with P = 2 A P=2A , find the largest possible value of A A .


The answer is 6.

Dexter Woo Teng Koon by Dexter Woo Teng Koon , 20, Malaysia

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2 solutions

Maria Kozlowska
Jun 27, 2017

It has to be a Heronian triangle with inradius r = A s = 1 r=\frac{A}{s}=1 . Using Exact formula for Heronian triangles for the inradius we get r = k ( m n − k 2 ) = 1 ⇒ k = 1 , m = 2 , n = 1 , a = 5 , b = 4 , c = 3 ⇒ A = 6 r=k(mn-k^2)=1 \Rightarrow k=1, m=2, n=1, a=5, b=4, c=3 \Rightarrow A=6 .

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Since sides are integers the perimeter too must be an integer. But P=2A so 2A also must be an integer. These conditions are only possible for a pythagorean triangle. But the only triangle with these condition is 3-4-5. So A=1/2 * 3 * 4=6.

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