Question

How to find the maximum area of a triangle if given the perimeter is 10 ?

Answer 1

Right triangle? <.<

Answer 2

didnt say it in the question :)

Answer 3

hmmmmmmmm

Answer 4

i tried suppose this triangle has maximum area if it is a right triangle, but this be more complicated for me :(

Answer 5

I was trying to use Heron's Formula,
but I can't seem to get it in terms of a single variable.

Or maybe using \(\Large\rm Area=a b\cos \theta\) where a and b are the legs around angle theta.

But that introduces an angle...... doesn't seem very helpful... hmm

Answer 6

Hmmm... I thnk you will be stuck with Heron's Formula.

Answer 7

That, and the Isoperimetric Property of Equilateral Triangles

Answer 8

I had to look up the name of it. LOL. I remembered that equilateral triangles had the largest area, but could not remember the name of the theorem.

That simplifies Heron's Formula down a lot:
\(A=\sqrt{s(s-a)(s-b)(s-c)} \text{ and } a=b=c \implies \)

\(A=\sqrt{s\left(s-\dfrac{2}{3}s\right )\left(s-\dfrac{2}{3}s\right )\left(s-\dfrac{2}{3}s\right )} \implies \)

\(A=\sqrt{3s\left(s-\dfrac{2}{3}s\right )} \implies \)

\(A=\sqrt{3s^2-2s} \)

And as always, \(s=\dfrac{a+b+c}{2}\)

Answer 9

pythagoras theorem?

Answer 10

Think of it like a rectangle... The maximum area triangle has to be an equilateral triangle, because all the inner angles being equal optimizes the area. It's the same way a rectangle with the highest area has to be a square.

Answer 11

In this case we don't know the triangle is a right triangle.
To maximize the area of any general shape, you want the sides and angles to be equal
So this triangle is an equilateral triangle.

Answer 12

okay, now be easy for mee to determine the area of this triangle if it is an equi triangle. thanks guys :)

Answer 13

for general case, you can set it up as an optimization problem :
Maximize : \(A(a, b, c) = \sqrt{ 5(5-a)(5-b)(5-c)} \)

subject to : \(a+b+c = 10\)

Answer 14

http://www.wolframalpha.com/input/?i=maximize+5%285-x%29%285-y%29%285-z%29%2C+x%2By%2Bz%3D10

Answer 15

can i hv a medal?

Answer 16

We know all the angle measurements are the same.
Since triangles add to 180 degrees we divide by 3 to find all the angles are 60 degrees
Here we're looking at side measure.
Each side is the same value, so I'll just say variable b equals the length of each side.
Each side is b units long. What is the total perimeter?