To solve for the area of the inner square tilted inside the outer square, let's first understand the configuration of the squares based on the given information.
The sides of the outer square are divided into two segments of lengths 8 units and 3 units. This means each side of the outer square has a total length of:
\[ 8 + 3 = 11 \text{ units} \]
Given that the outer square is a square, we can confirm that each side of the outer square measures 11 units.
Since the inner square is tilted inside the outer square, we can determine its dimensions by finding the vertical and horizontal distances it extends into the sides of the outer square.
To clarify the position of the inner square, let's denote the vertices of the outer square as \( A, B, C, D \) in a clockwise manner, where:
- \( AB = 11 \) units
- \( BC = 11 \) units
- \( CD = 11 \) units
- \( DA = 11 \) units
Assuming the inner square's vertices touch the midpoints of the outer square's sides, it will create segments along the outer square’s sides.
For each side of the outer square (let’s use side \( AB \) as an example):
- The segment from \( A \) to the intersection point with the inner square is \( 8 \) units.
- The segment from the intersection point to \( B \) is \( 3 \) units.
Thus, as the inner square intersects at the midpoints:
- The length of the side of the inner square can be derived from the distances it covers along the outer square’s sides.
Now, the distances from the corners of the inner square to the corners of the outer square mirror the side lengths:
- Each pair of adjacent segments (i.e., segments of lengths 8 and 3) show that the inner square effectively uses these lengths as its horizontal and vertical components.
This means, since the lengths split the sides evenly around the angles of the square, the length of the inner square can be calculated using the right triangles formed by the segments 8 and 3: \[ s = \sqrt{(8 + 3)^2 + (8 + 3)^2} = \sqrt{11^2 + 11^2} \]
However, since we only require the inner square’s area: The distance from the center vertically or horizontally across the square is critical in calculating the area. Each side of the inner square ratio: The inner square forms two right-angled triangles with legs measuring 8 and 3:
To find the inner square area denoted in terms of its side length \( s \):
Using the properties of \( s \): \[ \text{side length} = \sqrt{8^2 + 3^2} = \sqrt{64 + 9} = \sqrt{73} \]
The area of the inner square thus becomes: \[ A = (\sqrt{73})^2 = 73 \text{ square units} \]
Therefore, the area of the inner square is:
\[ \boxed{73} \text{ units}^2 \]
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