express the perimeter of triangle P as function of area A

**sluggerbroth** · May 2nd 2012, 01:50 PM

**emakarov** · May 2nd 2012, 01:58 PM

This is impossible. All triangles with vertices at (n, 0), (-n, 0) and (0, 1/n) have area 1.

**Soroban** · May 2nd 2012, 05:59 PM

Hello, sluggerbroth!

Who asked this question? . . . It is stupid!

Express the perimeter of triangle $P$ as function of area $A$ .

Code:

    - - - - * - - - - - - - - - - - - * - - - - - - - - - - - - * -
    :      * *                      * *                      * *  :
    :     *   *                   *   *                   *   *   :
    3    *     *                *     *                *     *    3
    :   *       *             *       *             *       *     :
    :  *         *          *         *          *         *      :
    - *  *  *  *  * - - - *  *  *  *  * - - - *  *  *  *  * - - - -
      : - - 4 - - :       : - - 4 - - :       : - - 4 - - :

All these triangles have an area of 6 square units.

But obviously they have different perimeters.

So $P \:=\:f(6)\,$ has at least three possible values.
. . It is not a function.

**MathoMan** · May 2nd 2012, 09:40 PM

How about $P=a+b+c$ and $A=\rho \cdot s$ where $\rho$ is the radius of the inscribed circle, and $s=\frac{a+b+c}{2}=\frac{P}{2}$ . Now $P=\frac{2 A}{\rho}$ . This can be considered a function in A (not just A, but A nevertheless).

**sluggerbroth** · May 3rd 2012, 02:42 AM

This comes from Schaum’s Outlines Precalculus
It is a two part question and reads as follows:
A) Express the area A of equilateral triangle as function of one side s.
B) Express the perimeter of the triangle P as a function of the area A.

The answers listed are:
A) A(s)=(s squared )(square root 3)/4
and
B) P(A)=(6 square root A)/ (fourth root 3)

How to get to these answers???? HELP!!

**emakarov** · May 3rd 2012, 04:53 AM

Originally Posted by sluggerbroth

A) Express the area A of equilateral triangle as function of one side s.

You mislead people in this thread by omitting the word "equilateral." To answer this question, draw an altitude, which is also a median, and use the Pythagorean theorem to find its length.

**sluggerbroth** · May 3rd 2012, 05:06 AM

I got part a of this but not part b B) Express the perimeter of the triangle P as a function of the area A.

How to get to this answer???? HELP!!

**emakarov** · May 3rd 2012, 05:12 AM

Originally Posted by sluggerbroth

I got part a of this but not part b B) Express the perimeter of the triangle P as a function of the area A.

So, P = 3s and you cannot express s through A from the equation $A=\frac{s^2\sqrt{3}}{4}$ ? Multiply both sides by $4/\sqrt{3}$ and takes the square root of both sides.

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