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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.Â
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.Â
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?Â
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.Â
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)Â
The area of a trapezoid isÂ
A = (b1 + b2) * h / 2
b1 = 6
b2 = 12Â
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.Â
Do you think I cooked the numbers. You should answer yes.Â
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.Â
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. Â
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.