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I have having trouble understanding how to break this problem apart.

I have an $ L$ shape with a rectangle in it.

The smaller rectangle has a side of $5 m$ and a side of $7 m$, the $L$ shape has an area of $6.25m^2$

I can work out the total area for both but not how to use the area to calculate the perimeter of the $L$ shape when no sides length or width are know.

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3 Answers

The geometric problem is not fully specified. We assume that the vertical and horizontal arms of the L have the same thickness $d$. (We were not told that the thicknesses are the same.) We suppose also that the $5\times 7$ rectangle referred to is the rectangle embraced by the two arms of the L, but outside the L.

Then if we draw the L shape, the little $d \times d$ square in the lower left corner has area $d^2$, and the rest of the L has area $d(5+7)$. So we obtain the equation $$d^2+12d=6.25.$$ Rewrite as $d^2+12d-6.25=0$, and solve. The positive root is $1/2$.

Now the perimeter of the L is easy to find. It is $2(7) +2(5)+4(1/2)$.

Remark: Is the above the right interpretation? That is certainly not clear. But the happy fact that the discriminant $(12)^2-(4)(-6.25)$ is a perfect square is evidence of sorts that the interpretation might be the intended one.

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I think that there is a lack of informations .

Let's denote perimeter of the shape as $P$ and total area as $A$ .

$A=(x+5)(y+7)-5\cdot 7$ (see picture below)

$P=2x+2y+24$

therefore , we have following two equations :

$\begin{cases} A=xy+7x+5y \\ P=2x+2y+24 \end{cases}$

hence , you cannot calculate value of perimeter exactly .

enter image description here

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added a graphic to better describe my problem –  Ashley Hughes Apr 7 '12 at 6:52
    
It is a question given by my Instructor, I assume that it is solvable. But I am not sure how to attack it at all –  Ashley Hughes Apr 7 '12 at 10:33
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up vote 0 down vote accepted

Ok so my mate Ryan showed me the way.

$A=lw$
$A=(5+w)(7+w)$
$A=(5+x)(7+x)$
$A=5*7+x*7+5*x+x*x$
$A=35+7x+5x+x^2$
$A=x^2+12x+35=41.25m^2$
That is the total area $A_F = A_T - A_S = 41.25 - 35 = 6.25$
$A=x_2+12x=6.25$
$A=x_2+12x-6.25=0$ Plug that into a grapher x = 0.5m

That gives me the sides and the question is answered.

Thanks for all the help from everyone it was more me not being able to express my question properly

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