To solve for $x$ in this circle geometry problem, let's analyze the given information.

Given:

1. The angle subtended by chord $AB$ at the center is $100^\circ$.
2. There are two inscribed angles, $5x^\circ$ and $3x^\circ$, formed by the same chord $AB$.

Solution:

In a circle, an inscribed angle that subtends the same arc as a central angle is half the measure of the central angle.

1. The central angle subtending arc $AB$ is $100^\circ$.
2. Therefore, the inscribed angle subtending arc $AB$ is: $\frac{100^\circ}{2} = 50^\circ$
3. This means: $5x + 3x = 50^\circ$
4. Simplify the equation: $8x = 50^\circ$
5. Solve for $x$: $x = \frac{50^\circ}{8} = 6.25^\circ$

Answer:

The value of $x$ is $6.25^\circ$.

Would you like more details on this solution or have any questions?

Related Questions:

1. What is the difference between central and inscribed angles in a circle?
2. How do you find the measure of an angle subtended by a chord in a circle?
3. What other properties are unique to inscribed angles in a circle?
4. Can inscribed angles be used to find arc lengths in the circle?
5. How would this problem change if the angle at the center was different?

Tip:

Remember that an inscribed angle is always half the measure of the central angle subtending the same arc.