The problem shows a circle with a central point $O$. There are points $U$, $S$, and $P$ on the circumference, and the angle $\angle USP = 35^\circ$ is inscribed in the circle. The goal is to find the angle $x$, which is $\angle UOP$, the central angle subtended by the arc $UP$.

Solution:

In a circle, the central angle is twice the inscribed angle that subtends the same arc. This is a known property of circles.

Given:

• Inscribed angle $\angle USP = 35^\circ$

We need to find:

• Central angle $x = \angle UOP$

Using the property: $x = 2 \times \angle USP = 2 \times 35^\circ = 70^\circ$

Answer:

The value of $x$ is $70^\circ$.

Would you like further details or have any questions?

Follow-up Questions:

1. What is the relationship between a central angle and an inscribed angle subtended by the same arc in a circle?
2. How would you find the length of the arc subtended by a central angle?
3. Can an inscribed angle be larger than its corresponding central angle?
4. How would the result change if $\angle USP$ were given as $40^\circ$?
5. What is the formula for finding the area of a sector based on the central angle?

Tip: Always remember that the central angle of a circle is twice the inscribed angle that subtends the same arc.