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Let's solve the problem step by step.

### Problem Recap:
- Two \(8 \times 8\) squares intersect at right angles, bisecting their intersecting sides.
- 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 \(16a - πb\).
- The goal is to find the value of \(a + b\).

### Step 1: Analyze the intersection of the squares
Each square has a side length of 8 units. Since the squares intersect at right angles, they form a cross-like shape. The points of intersection form a smaller square at the center, with sides of 8 units as well.

### Step 2: Determine the diameter of the circle
The diameter of the circle is the diagonal of the smaller square formed by the intersection. The side of this square is 8 units, so the diagonal \(d\) is given by:
\[
d = 8\sqrt{2}
\]
This is the diameter of the circle.

### Step 3: Calculate the area of the circle
The area \(A_{\text{circle}}\) of the circle with diameter \(d\) is:
\[
r = \frac{d}{2} = \frac{8\sqrt{2}}{2} = 4\sqrt{2}
\]
\[
A_{\text{circle}} = πr^2 = π(4\sqrt{2})^2 = 32π
\]

### Step 4: Calculate the area of the squares
Each square has an area of:
\[
A_{\text{square}} = 8 \times 8 = 64
\]
Since there are two squares:
\[
A_{\text{squares}} = 2 \times 64 = 128
\]

### Step 5: Determine the shaded area
The shaded area is the total area of the squares minus the area of the circle:
\[
\text{Shaded Area} = 128 - 32π
\]
This is given in the problem as \(16a - πb\).

### Step 6: Equate and solve for \(a\) and \(b\)
\[
16a - πb = 128 - 32π
\]
This implies:
\[
16a = 128 \quad \text{and} \quad b = 32
\]
Solving for \(a\):
\[
a = \frac{128}{16} = 8
\]
So, \(a = 8\) and \(b = 32\).

### Step 7: Calculate \(a + b\)
\[
a + b = 8 + 32 = 40
\]
Thus, the correct answer is **not listed** in the options, which suggests there might be an issue with either the problem's options or the derivation. 

Would you like to explore this problem further or focus on other aspects?

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### Relative Questions:
1. How do you derive the area of a circle from its radius?
2. What is the significance of the diagonal in determining the circle's diameter in this problem?
3. Can you calculate the area of a circle when given only the circumference?
4. How do intersections of geometric shapes typically affect area calculations?
5. What are some other methods to find areas of composite shapes like the one in this problem?

### Tip:
When solving geometry problems involving composite shapes, it often helps to break down the problem into simpler shapes (like squares, circles, or triangles) and analyze their relationships carefully.

Two 8 × 8 squares intersect at right angles, bisecting their intersecting sides. 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 16a - πb. The value of a + b is?

Explore how to calculate the value of a + b in a geometry problem involving two intersecting 8×8 squares and a circle. This problem covers key geometric concepts, including the area of a circle and squares, and the significance of intersecting shapes. Suitable for students in Grades 9-10.

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