explain to a schook kid on how to solve this question.---**Question Stem:**
34. Noor drew a small square. Using its diagonal, she drew a larger square. Using the diagonal of this new square, she drew another square. She repeated this again to draw the largest square seen in the shape below.
**Chart/Diagram Description:**
* **Type:** Geometric figure, specifically a series of nested squares.
* **Main Elements:**
* There are four squares, nested within each other, with their orientations alternating.
* **Smallest Square (Square 1):** Located at the bottom-left corner of the next square. It is oriented with its sides horizontal and vertical. It is shaded with a dense, dark hatching pattern. Its diagonals are drawn, intersecting at its center. The number '4' is written near two of its adjacent sides, indicating its side length is 4 units.
* **Second Square (Square 2):** This square is drawn using the diagonal of the smallest square. Its vertices appear to coincide with the midpoints of the sides of the third square. It is rotated 45 degrees relative to the smallest square. It is unshaded. Its diagonals are also drawn, intersecting at its center.
* **Third Square (Square 3):** This square is drawn using the diagonal of the second square. It is oriented with its sides horizontal and vertical. It is unshaded. Its vertices appear to coincide with the midpoints of the sides of the largest square.
* **Largest Square (Square 4):** This square is drawn using the diagonal of the third square. It is the outermost square in the figure and is rotated 45 degrees relative to the third square. It is shaded with a lighter, less dense hatching pattern.
* **Relative Position:** The squares are centered around a common point (the center of the largest square). Each subsequent square's diagonal is used as the side length of the next larger square, and their orientations alternate by 45 degrees. The smallest square is placed such that one of its vertices is at the center of the third square, and two of its sides form part of the boundary of the third square.
**Problem Statement:**
The area of the smallest square is 16 cm².
What is the area of the whole shape, in cm²?
**Options:**
(A) 128
(B) 184
(C) 240
(D) 256
**Other Relevant Text:**
* Page number: 9
* Source/Copyright: 2010 ICAS Mathematics Paper C EAA
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Let's understand this step by step. Noor drew four nested squares. The smallest square has an area of 16 square centimeters, which means its side length is 4 units. Each new square is created by using the diagonal of the previous square as the side length of the next square. Our goal is to find the total area of all four squares combined.
Now let's understand how the diagonal relates to the side length. The smallest square has a side length of 4 units. To find its diagonal, we use the Pythagorean theorem. Since it's a square, both sides are equal to 4. So we have 4 squared plus 4 squared equals diagonal squared. That's 16 plus 16 equals 32. Taking the square root, the diagonal equals 4 times the square root of 2, which is approximately 5.66 units. This diagonal becomes the side length of the next larger square.
Now let's find all the side lengths systematically. We start with Square 1, which has a side length of 4. Its diagonal becomes the side of Square 2, which is 4 times square root of 2. The diagonal of Square 2 becomes the side of Square 3. Since 4 square root of 2 times square root of 2 equals 8, Square 3 has a side length of 8. Finally, Square 4 has a side length equal to the diagonal of Square 3, which is 8 times square root of 2. Notice the pattern: each side length is multiplied by square root of 2 to get the next one.
Now let's calculate the area of each square using the formula Area equals side squared. Square 1 has area 4 squared equals 16 square centimeters. Square 2 has area 4 square root of 2 squared, which equals 32 square centimeters. Square 3 has area 8 squared equals 64 square centimeters. Square 4 has area 8 square root of 2 squared equals 128 square centimeters. Notice an interesting pattern: each area is exactly double the previous one! This doubling pattern will help us verify our final answer.
Finally, let's add all four areas together to find the total area of the whole shape. We have 16 plus 32 plus 64 plus 128. Let's calculate step by step: 16 plus 32 equals 48. Then 48 plus 64 equals 112. Finally, 112 plus 128 equals 240. Therefore, the total area of the whole shape is 240 square centimeters. Looking at our answer choices, this matches option C. The answer is 240 square centimeters.