Each of the sides of a square (S) with an area of 16 is bisected and a smaller square (S2) is constructed using the bisected parts as vertices. The same process is carried out on S2 to construct an even smaller square (S3). What's the area of S3? (Draw a diagram, please?)
The square S is cut into 5 pieces by S2, namely the square S2, and 4 triangles at each of the four corners.
If you flip each of the four triangles at the four corners about the sides of S2 so they go inside of S2, you will find that the sum of the areas of the four triangles equal to the area of S2.
In other words, the area of S2 is half that of S (16/2=8).
Similarly, S3's area is half that of S2 (8/2=4), and so on.
Draw a diagram, or even better, cut out a square piece of paper and demonstrate the above.
To find the area of S3, we can follow the steps below:
Step 1: Start with the given square S with an area of 16.
-- S is a square with four sides of equal length.
-- The length of each side of S is √16 = 4 units.
Step 2: Bisect each side of S to form the smaller square S2.
-- Each side of S2 will have a length equal to half of S, which is 4 / 2 = 2 units.
-- S2 will have four sides of length 2 units.
Step 3: Construct the even smaller square S3 using S2.
-- Bisect each side of S2 to form S3.
-- Each side of S3 will have a length equal to half of S2, which is 2 / 2 = 1 unit.
-- S3 will have four sides of length 1 unit.
Step 4: Calculate the area of S3.
-- The area of a square is given by side length squared.
-- The area of S3 = side length of S3 squared.
-- The area of S3 = 1^2 = 1 square unit.
Therefore, the area of S3 is 1 square unit.
Please refer to the diagram here:
```
_______
| |
| S3 |
|_______|
_________ _______ _______
| | | | | |
| S | ---> | S2 | ---> | S3 |
|_______| |_____| |_____|
_______ _______
| | | |
| S | | S2 |
|_______| |_____|
```
Note: The diagram shows the three squares, S, S2, and S3, after each step.
To find the area of the smaller square S3, let's start by visualizing the process step by step.
Step 1: Start with the original square S with an area of 16.
Step 2: Each side of S is bisected, creating 4 smaller squares, let's call them S2a, S2b, S2c, and S2d. Each smaller square has a side length equal to half the side length of the original square S.
Step 3: Construct a square S2 using the vertices of the smaller squares S2a, S2b, S2c, and S2d. This square S2 will have a side length equal to the sum of the side lengths of the smaller squares S2a, S2b, S2c, and S2d.
Step 4: Repeat the process on S2 to construct an even smaller square S3.
Now, let's calculate the side length of S2:
The original square S has an area of 16, so its side length is √16 = 4.
Each side of S is bisected, resulting in smaller squares S2a, S2b, S2c, and S2d. Since S2a is formed by bisecting one side of S, its side length is half of the side length of S, which is 4/2 = 2. Therefore, the side length of S2 is the sum of the side lengths of S2a, S2b, S2c, and S2d, which is 2 + 2 + 2 + 2 = 8.
Now, let's calculate the side length of S3:
Using the same process, S3 is constructed using the vertices of the smaller squares S3a, S3b, S3c, and S3d (resulting from bisecting the sides of S2). Since S2 has a side length of 8, each smaller square S3a, S3b, S3c, and S3d will have a side length of 8/2 = 4.
Therefore, the side length of S3 is the sum of the side lengths of S3a, S3b, S3c, and S3d, which is 4 + 4 + 4 + 4 = 16.
Hence, the area of S3 is (side length of S3)² = 16² = 256.
So, the area of S3 is 256 square units.
(Note: Unfortunately, I cannot draw a diagram as a text-based AI, but I hope the explanation helps you visualize the process!)