Rectangles
The ratio of the length to the width of one rectangle is proportional to the ratio of the length to the width of a smaller rectangle. Describe the circumstances for which this proportion involves a geometric mean.
if x is a geometric mean between two numbers a and b, then
x/a = b/x
x^2 = ab
x = √(ab)
if the length of the small one is l
and its width is w
then the width of the larger has to be l
and the length has to be l2
so l/w = l2/w
w = √(l*l2)
e.g. small rectangle is 4 by 6, larger is 6 by 9
notice 4/6 = 6/9
and 6^2 = 4*9
e.g. first rectangle
6/4 = 9/6 6^2 =36
In order for the proportion of the length to the width of one rectangle to involve a geometric mean, the following circumstances need to be met:
1. The two rectangles must be similar: In other words, their corresponding angles are equal, and their corresponding sides are proportional.
2. The ratio of the length to the width of the larger rectangle must be equal to the ratio of the length to the width of the smaller rectangle.
When these conditions are met, the proportion between the length and width of the two rectangles becomes a geometric mean. This means that the length of the larger rectangle divided by its width is equal to the length of the smaller rectangle divided by its width. In mathematical terms, if the length of the larger rectangle is L1, the width of the larger rectangle is W1, the length of the smaller rectangle is L2, and the width of the smaller rectangle is W2, then the proportion involving a geometric mean can be written as:
L1/W1 = L2/W2
It's important to note that this proportion involves a geometric mean because the lengths and widths of the rectangles are being compared and related to each other in a way that preserves the similarity between the rectangles.
To determine under what circumstances the ratio of the length to the width of one rectangle is proportional to the ratio of the length to the width of a smaller rectangle involving the geometric mean, we need to use the concept of similar rectangles.
Two rectangles are considered similar if their corresponding angles are congruent and their sides are proportional. In your case, the ratio of the length to the width of the larger rectangle is proportional to the ratio of the length to the width of the smaller rectangle.
Let's denote the length of the larger rectangle as L1, the width of the larger rectangle as W1, the length of the smaller rectangle as L2, and the width of the smaller rectangle as W2.
The ratio of the length to the width of the larger rectangle is L1/W1, and the ratio of the length to the width of the smaller rectangle is L2/W2. The given condition is that these ratios are proportional.
In order for this proportion to involve the geometric mean, we need to have:
L1/W1 = (L2/W2)^k
Where k is a constant and the geometric mean can be found within this equation.
To explain this relationship further, the geometric mean is a type of average that is calculated by taking the square root of the product of two values. In this case, if the ratio of the length to the width of the larger rectangle is equal to the geometric mean of the ratio of the length to the width of the smaller rectangle, then it satisfies this condition.
In summary, the circumstances for which this proportion involves a geometric mean is when the ratio of the length to the width of the larger rectangle is equal to the geometric mean of the ratio of the length to the width of the smaller rectangle, raised to some constant power k.