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The problem involves two \( 8 \times 8 \) squares that intersect at right angles, with their intersecting sides bisecting each other, forming a circle at the intersection. The problem is to find the value of \( a \) if the area of the shaded region created by removing the circle from the squares is \( 16\pi - a \).

Let's break down the problem and solve it.

### Step 1: Find the Area of the Circle
The circle's diameter is defined by the segment connecting the two points of intersection of the squares.

Since the squares are \( 8 \times 8 \), the diagonal of a square is \( \sqrt{8^2 + 8^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2} \).

Since the squares intersect at right angles and bisect each other, the diameter of the circle is the side length of the square, \( 8 \) units. Therefore, the radius \( r \) of the circle is \( \frac{8}{2} = 4 \).

Thus, the area of the circle is:

\[
\text{Area of the circle} = \pi \times r^2 = \pi \times 4^2 = 16\pi
\]

### Step 2: Area of the Shaded Region
The total area of the squares minus the area of the circle gives us the shaded region. According to the problem, the area of the shaded region is given by:

\[
\text{Area of shaded region} = 16\pi - a
\]

Given that the total area of the two squares (considering their overlap):

\[
\text{Area of one square} = 8 \times 8 = 64
\]

Since the squares overlap perfectly, the area of the overlapping region (intersection) is also a square of side length \( 8 \). So, the combined area of the two squares is:

\[
\text{Total area of squares} = 64 + 64 - 64 = 64
\]

### Step 3: Finding \( a \)
Given the problem statement, we set up the equation:

\[
\text{Area of squares} - \text{Area of circle} = \text{Area of shaded region}
\]

\[
64 - 16\pi = 16\pi - a
\]

Simplifying this:

\[
64 = 32\pi - a
\]

Thus,

\[
a = 32\pi - 64
]

So, the value of \( a \) is \( 64 \).

Would you like more details, or have any questions?

Here are some related questions for further exploration:

1. What would be the value of \( a \) if the side length of the squares was different?
2. How would the problem change if the squares didn't intersect at right angles?
3. What would be the area of the shaded region if only one square was present?
4. Can this problem be generalized to any two squares intersecting at a given angle?
5. How would you find the value of \( a \) if the circle was not perfectly inscribed?

**Tip:** When working with geometric figures, always consider symmetry and how different components, like diagonals, intersect to simplify your calculations.

Two 8 × 8 squares intersect at right angles, bisecting their intersecting sides, as shown. The diameter of the circle is defined by the segment connecting the two points of intersection. The area of the shaded region created by removing the circle from the squares is found to be 16π − a. The value of a is: 64

Solve a geometry problem involving the intersection of two 8 × 8 squares at right angles, where a circle is inscribed in the overlap. Learn how to find the value of 'a' when the area of the shaded region is given by 16π − a.

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