PixelForge, a revolutionary graphics startup, is developing a new algorithm to generate increasingly intricate fractal patterns for their immersive virtual reality environments. Their core design principle starts with a perfect square on a digital canvas. For each subsequent iteration, they find the midpoints of the sides of the current square and connect these midpoints to form a new, smaller square inside the previous one. They repeat this process. If the very first outermost square generated by PixelForge's algorithm has an area of 100 square units, what is the perimeter of the *third* square formed in this iterative process (i.e., the second square *inside* the original one)?
Correct: 20 units
Let's denote the squares as S1 (outermost), S2 (first inner), and S3 (second inner, which is the 'third' square in the sequence as per the question).
1. **Square S1 (Original):**
Area = 100 square units.
Since Area = side * side, the side length of S1 is `sqrt(100) = 10` units.
2. **Square S2 (First Inner Square):**
This square is formed by connecting the midpoints of S1's sides. A fundamental geometric property (which can be proven using the Pythagorean theorem) is that when you connect the midpoints of a square's sides to form a new square, the new square has exactly half the area of the original square.
So, Area of S2 = Area of S1 / 2 = 100 / 2 = 50 square units.
(To confirm: if S1 has vertices at (0,0), (10,0), (10,10), (0,10), its midpoints are (5,0), (10,5), (5,10), (0,5). The side length of S2 would be `sqrt((5-0)^2 + (0-5)^2) = sqrt(25 + 25) = sqrt(50)` units. Area = `(sqrt(50))^2 = 50`.)
3. **Square S3 (Second Inner Square / 'Third' Square in the sequence):**
This square is formed by connecting the midpoints of S2's sides. Following the same property, its area will be half that of S2.
Area of S3 = Area of S2 / 2 = 50 / 2 = 25 square units.
Now, we need to find the side length of S3: `side_S3 = sqrt(25) = 5` units.
The question asks for the perimeter of this third square.
Perimeter of S3 = `4 * side_S3 = 4 * 5 = 20` units.