To solve this problem, we need to find the area of the portion of the circle that lies within the triangle $ABC$, where:

• $A = (0, 0)$,
• $B = (6, 0)$,
• $\angle A = 60^\circ$,
• $\angle B = 45^\circ$,
• The circle has radius $r = \dfrac{3\sqrt{2}}{2}$ and is centered at the origin.

Step 1: Understand the Geometry

The triangle has vertex $A$ at the origin $(0, 0)$ and vertex $B$ on the x-axis at $(6, 0)$.

Using the given angles at $A$ and $B$:

• $\angle A = 60^\circ$,
• $\angle B = 45^\circ$.

We can determine that the remaining angle at $C$ is: $\angle C = 180^\circ - 60^\circ - 45^\circ = 75^\circ.$

Step 2: Locate Point $C$

Since we know two angles and the length of one side (the base $AB = 6$), we can use the Law of Sines to find the other sides and locate point $C$.

Using the Law of Sines: $\frac{AB}{\sin C} = \frac{BC}{\sin A} = \frac{AC}{\sin B},$ we calculate $BC$ and $AC$ (the distances of the remaining sides).

For now, let's calculate the side $BC$: $\frac{6}{\sin 75^\circ} = \frac{BC}{\sin 60^\circ},$ which gives: $BC = \frac{6 \cdot \sin 60^\circ}{\sin 75^\circ}.$ After solving, $BC \approx 5.21$.

Similarly, we can find $AC$.

Step 3: Find the Area of the Circle Within the Triangle

The area of the portion of the circle within the triangle depends on the angles formed at the origin and the intersection of the triangle with the circle. To accurately determine the area, we would need to compute the sector of the circle that lies inside the triangle.

Given that the circle is centered at the origin and has a radius $r = \frac{3\sqrt{2}}{2}$, we now compute the portion that falls within the boundaries of the triangle using appropriate geometric formulas or integration methods (in polar coordinates).

Since the setup of the geometry is intricate, the final result will involve subtracting unwanted parts (those outside the triangle) from the full circular sector.

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Would you like me to clarify any specific step in detail or work through the geometry and final integration more explicitly?

Here are some questions to extend understanding:

1. What is the Law of Sines, and how can it be applied to solve triangle problems?
2. How do you calculate the area of a sector of a circle?
3. Why does the sum of the angles in a triangle always equal 180°?
4. How does knowing two angles help determine the third in a triangle?
5. What role does polar coordinates play in integrating areas?

Tip: When working with triangles, always try to identify right angles or known angle relationships to simplify calculations.