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Problem Description:

1. Two circles $C_1$ and $C_2$ with equal radii $r$ intersect at points $A$ and $B$, forming a lens-shaped shaded region.
2. The line segment $C_1C_2$ is the distance between the centers of the circles, and it's given that $C_1C_2 = r$.
3. In part (a), we are told that the perimeter of the shaded region is $4\pi \, \text{cm}$, and we are to find the radius $r$.
4. Part (b) asks us to calculate the exact area of the shaded region.

(a) Perimeter of the Shaded Region:

The shaded region's perimeter is composed of two circular arcs:

1. Arc from $A$ to $B$ on $C_1$'s circumference.
2. Arc from $A$ to $B$ on $C_2$'s circumference.

The total perimeter of the shaded region is $4\pi \, \text{cm}$, and we know that:

• Each arc is a semicircle because $C_1C_2 = r$. Thus, the two arcs together form a full circle.

The perimeter of one full circle is $2\pi r$. Setting this equal to the given total perimeter: $2\pi r = 4\pi$ Dividing through by $2\pi$: $r = 2 \, \text{cm}$

(b) Area of the Shaded Region:

The shaded region is the intersection of two circles. To find the area of this lens-shaped region, we subtract the area of the two triangular segments from the area of the overlapping circular segment.

1. The two intersecting circles form two circular segments, each subtending an angle of $\frac{\pi}{3}$ radians (60°) at the circle's center, since $C_1C_2 = r$ and forms an equilateral triangle with radii.

2. Area of one circular segment: $\text{Area of segment} = \frac{1}{2} r^2 (\theta - \sin\theta)$ Here $\theta = \frac{\pi}{3}$. Using $r = 2$: $\text{Area of one segment} = \frac{1}{2} (2^2) \left(\frac{\pi}{3} - \sin\frac{\pi}{3}\right)$ $= 2 \left(\frac{\pi}{3} - \frac{\sqrt{3}}{2}\right)$

3. Total shaded area: The shaded region consists of two circular segments, so: $\text{Shaded area} = 2 \times \text{Area of one segment}$ $\text{Shaded area} = 2 \cdot 2 \left(\frac{\pi}{3} - \frac{\sqrt{3}}{2}\right)$ $\text{Shaded area} = 4\left(\frac{\pi}{3} - \frac{\sqrt{3}}{2}\right) \, \text{cm}^2$

Final Answers:

(a) $r = 2 \, \text{cm}$
(b) $\text{Exact Area} = 4\left(\frac{\pi}{3} - \frac{\sqrt{3}}{2}\right) \, \text{cm}^2$

Would you like me to explain any part further?
Here are some related questions for practice:

1. How do you derive the formula for the area of a circular segment?
2. If the distance between the circle centers were $r/2$, how would the problem change?
3. What is the geometric significance of the perimeter and area relationships?
4. Can you find the total area of the two circles, including the non-shaded parts?
5. What happens to the shaded region if the distance $C_1C_2 > r$?

Tip: Always visualize problems involving geometry and intersections to ensure clarity in the solution process!