Area and Perimeter

[Butter]

Draw a quadrilateral with an area of 20cm2 and a perimeter of 20cm.
I tried a couple but i couldn't make it. Can someone help please? But don't make it complicated, make it understandable.

 

[MarkFL]

Hint: Consider two 3-4-5 right triangles, and a rectangle put together to form an isosceles trapezoid.

 

[soroban]

Hello, Butter!

Draw a quadrilateral with an area of 20 cm² and a perimeter of 20 cm.

How about a rectangle?

            x
      *-----------*
      |           |
    y |           | y
      |           |
      *-----------*
            x

Area: .\(\displaystyle xy \:=\:20\) .[1]

Perimeter: .\(\displaystyle 2x + 2y \:=\:20 \quad\Rightarrow\quad y \:=\:10 - x\) .[2]

Substitute [2] into [1]: .\(\displaystyle x(10-x) \:=\:20 \quad\Rightarrow\quad x^2 - 10x + 20 \:=\:0\)

Quadratic Formula: .\(\displaystyle x \;=\;\dfrac{10 \pm\sqrt{10^2 - 4(1)(20)}}{2(1)} \;=\;\dfrac{10\pm\sqrt{20}}{2} \;=\;5 \pm\sqrt{5}\)

Therefore: .\(\displaystyle \begin{Bmatrix}x \;=\;5 + \sqrt{5} \\ y \;=\;5-\sqrt{5} \end{Bmatrix} \)


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MarkFL has an elegant solution.

He could also have formed a parallelogram.

               2     3
            *-----*-----*
           /:     :    /
          / :     :   /
       5 /  :4   4:  / 5
        /   :     : /
       /    :     :/
      *-----*-----*
         3     2