Maximum area of a quadrilateral

Geometry Level 3

What is the maximum area of a quadrilateral with sides 1, 4, 7, and 8?

Note: The figure shown is not drawn to scale or configuration.

1 solution

Ossama Ismail
Feb 14, 2017

It is the area of the two triangles $(1,8, \sqrt 65)$ and $(4,7, \sqrt{65})$ = $\frac{1 \times 8}{2} + \frac{4 \times 7}{2} = 18$ .

note that $1^2 + 8^2 = 4^2 + 7^2 = 65$

Another way to work this problem out is to make note of Brahmagupta's formula because concyclic quadrilateral has maximum area.

Michael Huang - 4 years, 3 months ago

Indeed. But could you prove this fact? @Michael Huang

Swapnil Das - 4 years, 3 months ago

NO. I don't have a prove !!

But here we have special case where the shape is constructed from two right angle triangles.

Ossama Ismail - 4 years, 3 months ago

@Ossama Ismail – Oh, I actually asked Michael for a Proof. Anyway, a nice solution.

Swapnil Das - 4 years, 3 months ago