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The problem involves two intersecting circles with the centers \( X \) and \( Y \), where the radius of the circle centered at \( X \) is 5 cm and that at \( Y \) is 3 cm. The circles intersect at points \( A \) and \( B \). It is also given that the angle \( \angle AYB = \frac{\pi}{3} \) radians.

The questions to solve are:
1. (a) Find \( \angle AXB \).
2. (b) Calculate the area of the shaded region.

### Part (a): Finding \( \angle AXB \)

Since we are given \( \angle AYB = \frac{\pi}{3} \) radians, we can use the properties of intersecting circles. Specifically, the points \( A \), \( X \), \( Y \), and \( B \) form a quadrilateral. In a cyclic quadrilateral (one inscribed in a circle), opposite angles add up to \( \pi \) radians.

Therefore:
\[
\angle AXB + \angle AYB = \pi
\]
\[
\angle AXB = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \, \text{radians}.
\]

### Part (b): Area of the Shaded Region

The shaded region is the common area between the two circles, which can be found by calculating the area of the circular segments formed by the intersecting points.

The total area of the shaded region is given by:
\[
\text{Area of the shaded region} = \text{Area of sector of circle X} - \text{Area of triangle XAB} + \text{Area of sector of circle Y} - \text{Area of triangle YAB}.
\]
We can break this calculation down into steps, considering that the angle subtended by the arc in both circles is \( \frac{\pi}{3} \) radians for circle \( Y \).

Let me calculate the areas of the two segments next.It seems like I can’t do more advanced data analysis right now. Please try again later.

For now, I can explain the rest of the steps conceptually for calculating the area of the shaded region, if you'd like. Let me know if you want further clarification or help on specific parts.

Here are 5 related questions to deepen your understanding:
1. How are angles in cyclic quadrilaterals related?
2. What is the formula for the area of a circular segment?
3. How does the angle in radians affect the area of a sector?
4. What method would you use to calculate the area of a triangle given three sides?
5. How does the intersection of two circles affect the common area between them?

**Tip:** Always express angles in radians when working with circular sectors and segments for precise calculations.

In the diagram, X and Y are centers of circles with radii 5 cm and 3 cm respectively. The two circles intersect at points A and B. Given that ∠AYB = π/3 radians, calculate (a) ∠AXB, (b) the area of the shaded region.

Solve a geometry problem involving two intersecting circles with centers X and Y, radii 5 cm and 3 cm, and an angle ∠AYB = π/3 radians. Learn how to calculate ∠AXB and the area of the shaded region using cyclic quadrilateral and circle sector formulas. Suitable for students in Grades 10-12.

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