SOLUTION: 5a. The sides of a square are 16 cm in length. The midpoints of the sides of this square are joined to form a new square and four triangles. The process is repeated twice. Let Xn

Question 1097196: 5a. The sides of a square are 16 cm in length. The midpoints of the sides of this square are joined to form a new square and four triangles. The process is repeated twice.
Let Xn denote the length of one of the equal sides of each new triangle.
Let An denote the area of each new triangle.
The following table gives the values of Xn and An, for 1 less than or equal to n less than or equal to 3.
N 1 2 3
Xn 8 __ 4
An 32 16 __

5b. The process described above is repeated. Find A6.

5c. Consider an initial square of side length k cm. The process describes above is repeated indefinitely. The total area of the shaded regions is k cm^2. Find the value of k.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

First note that the triangles are always isosceles right triangles; that means we will always have

It should be clear that X1 is half the side length of the original square, or 8.

So and

Now the side length of the second square is the hypotenuse of the isosceles right triangle with side length X1; that hypotenuse length is X1*sqrt(2), which is 8*sqrt(2). So then X2 is half that length, or 4*sqrt(2).

So and

Since the same process is being repeated over and over, it should be clear that the ratio of the Xi from one level to the next should always be the same, and likewise for the ratio of the Ai.

From the values we know for X1 and X2, we know the side length decreases by a factor of sqrt(2) each time; and from the values we know for A1 and A2, we know the areas decrease by a factor of 2 each time.

As a side note, there is a very powerful general principle regarding similar figures that can be demonstrated here. The general principle is that if the scale factor (ratio of linear measurements) between two similar figures is a:b, then the ratio of area measurements between the two figures is a^2:b^2. And in this problem we indeed see that the scale factor between successive triangles is sqrt(2):1, and the ratio of areas between successive triangles is the square of that, 2:1.

And now back to the problem....

We know the values of X1 and X2, and A1 and A2; and we know the factors by which the Xi and Ai decrease from one level to the next. So it it now easy to answer parts a and b of the problem:

For part c, there is no definition of which parts of the figure are shaded. However, it would make sense that the shaded regions are the triangles. And if the process is repeated indefinitely, then the whole original square will be shaded.

In that case, part c is asking for the side length "k" of a square such that the side length in cm is equal to the area in square cm. That's an easy question:

k = 0 or k = 1

The solution k=0 is not very interesting for this problem; so the answer to part c is k=1.

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