50 points! find the scale factor and ratio of perimeters for a pair of similar trapezoids with areas 81 cm squared and 36 cm squared

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What does the ratio mean when you take the two areas and compare them?
It's much easier if I begin by explaining what will happen in the case of a rectangle or a parallelogram. It's harder to see with a trapezoid because it has a unique formula. But believe it or not, it actually is the same as the other two for the purposes of this question.

Area of a rectangle.
A = L * W
L = 9 cm
W = 4 cm The units are very important.
Area = 9 cm * 4cm
Area = 36 cm^2

Now suppose you have a bigger rectangle
L = 13.5 cm
W = 6 cm
Area = L * W
Area = 13.5 cm * 6 cm
Area = 81 cm^2
Well look at that? Do you think I "cooked" those numbers? You'd be right
if you said I did.

What's the point? the point is that each dimension used to find the areas (L and W) were multiplied by 1.5 which gave the correct ratios. In other words, both L and W were multiplied by 1.5 to get  the relationship between the two rectangles' areas. 

Where did I get the 1.5
The 1.5 came from the sqrt(81/36) = 9/6 = 3/2 = 1.5

Why did I take the square root?
That's a little harder to explain and it's much harder for the trapezoid. So read on. 

The reason I took the square root is because I needed to adjust not one factor but two. Area is essentially in 2 dimensions L and W. They both have to be adjusted. That explains how much bigger the large rectangle is than the small one in terms of area. But what about their perimeters? 

P = 2(L + W) for the small rectangle
P1 = 2 (1.5L + 1.5W) for the large rectangle.
Take out the common factor of 1.5
P1 = 2*1.5 * (L + W)
P1 = 3 (L + W)

If you've followed me thus far what you should be suspecting is that the Perimeters are in the ratio of 3/2

Area they?
P_small rectangle = 2(9 + 4) = 26
P_large rectangle = 2(13.5 + 6) = 18.5 * 2 = 39

Ratio Large to Small = 39 / 26 = 3/2 (divide top and bottom by 13).

Believe it or not, I have just given you the answer to your problem about the trapezoids. Their dimensions will bear exactly the same relationship as these two rectangles. There is a wonderful math text out called crossing the river with dogs. It spends quite a bit of time discussing the solution to simple examples of very complex problems. That's what we have done here. It may not seem so, but we've solved a much simpler problem to get the answer to a very nasty one. 

And now for a discussion for the Trapezoids.
For all it's complexity the trapezoid is going to give you exactly the same kind of result (1.5) 

The area of a trapezoid is 
A = (b1 + b2) * h / 2
b1 = 6
b2 = 12 
h = 4
A_small = (6 + 12)*4 / 2
A_small = 18 * 4 / 2
A_small = 36

Now To get the large trapezoid, we multiply each of the dimensions by 1.5
b1 = 9
b2 = 18
h = 6
Area_larger = (9 + 18)*6/2 = 27 * 3 = 81. 
Do you think I cooked the numbers. You should answer yes. 

What about the 2 sides to these two trapezoids? You want them to be equal and they are. They calculate out to 5 for the small trapezoid and 7.5 for the large trapezoid which is about as lucky as it could be. 
The perimeter of the small trapezoid is
P_small = 5 + 5 + 6 + 12 = 28
P_Large = 7.5 + 7.5 + 9 + 18 = 42. Are these in the right ratio? Is the ratio 3/2
P_Large / P_ Small = 42/28 = 3/2 when top and bottom are divided by 14.  

So what's the answer?
The answer is take the square root of the area ratios. Reduced the result and that is the relationship between the 2 perimeters. I'm going to post this. I'll put the language in a comment. It would be horrifying if I lost all this.