In the figure above, the circle has center O. If the measure of arc APB is 56°, what is the value of x?
The value of x is 56°.
The measure of an arc in a circle is equal to the angle formed at the center of the circle that subtends that arc. In this case, since arc APB measures 56°, the angle x at the center O that subtends this arc must also be 56°.
This choice suggests that the angle at center O is half the measure of the arc, which is incorrect. The central angle is equal to the measure of the arc, not half of it. Therefore, 28° does not represent the correct relationship.
Similar to choice A, this option incorrectly assumes a different relationship between the arc and the angle. The angle x must be equal to the arc's measure of 56°, so 34° is not applicable in this scenario.
This is the correct answer. The measure of arc APB is directly equal to the angle x at the center O, which is also 56°. This direct relationship between the arc measure and the central angle confirms that x must be 56°.
This choice mistakenly suggests that the angle at the center is greater than the measure of the arc, which contradicts the established property of circles. Since the angle x corresponds exactly to the arc APB, 62° is not a valid measurement for angle x.
In a circle, the measure of an arc is equal to the angle formed at the center of the circle that subtends that arc. Thus, with arc APB measuring 56°, the angle x at center O must also measure 56°. The other options incorrectly interpret this relationship, confirming that x's value is indeed 56°.
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