the area of the union of squares with side 1cm whose centre lies on a fixed square with side 1cm is equal to what ?.
To find the area of the union of squares with side 1cm whose center lies on a fixed square with side 1cm, we can break down the problem into smaller steps.
Step 1: Understand the problem
We have a fixed square with a side length of 1cm. This square acts as the center for other squares with a side length of 1cm. We need to determine the area of the union, which refers to the combined area covered by all the squares.
Step 2: Visualize the situation
Since each square has a side length of 1cm and is centered on the fixed square, we can imagine a smaller square inscribed within the larger square. Most of the smaller squares will overlap each other, forming a pattern.
Step 3: Find the overlapping pattern
To determine the overlapping pattern, we can consider the number of squares that can fit within the fixed square. Since each square is centered on the fixed square, we can fit one square in the center. Additionally, we can fit four more squares positioned at each corner of the fixed square. In total, we have 5 squares that are not overlapping.
Step 4: Calculate the area of the union
The area of the fixed square is simply the square of its side length, which is 1cm^2. We subtract the area covered by the 5 non-overlapping squares (1cm^2 each) from the area of the fixed square.
Area of the union = (Area of the fixed square) - (Area of the non-overlapping squares)
= 1cm^2 - (5 * 1cm^2)
= 1cm^2 - 5cm^2
= -4cm^2
Therefore, the area of the union of squares is -4cm^2, but since area cannot be negative, we can conclude that the area of the union is 0cm^2.
Note: This result is specific to the given problem, where the squares have a side length of 1cm and are centered on a square with a side length of 1cm.