MHB Can you find x using the trigonometry of circle sectors?

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To find x in the circular sector problem, the arc-length formula \( s = r\theta \) can be applied to derive two equations involving x and the angle \( \theta \). Additionally, the concept of similarity can be utilized to establish a ratio between the radius and arc-length for the sectors. This approach allows for the formulation of equations that can help solve for the unknown variable x. Both methods provide a structured way to tackle the problem using trigonometric principles. Understanding these relationships is key to finding the solution effectively.
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I need help in solving this problem. Below shows all the measurements of the diagram, I need to find x:View attachment 6590
 

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For a circular sector of radius $r$, subtending angle $\theta$, the arc-length $s$ is given by:

$$s=r\theta$$

Can you apply this formula to get two equations in $x$ and $\theta$?

Or, we can use similarity to equate the ratio of radius to arc-length for both sectors. ;)
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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