Question

in the figure, point b lies on a circle centered at point p, and $overline{ac}$ is a diameter of the circle. not drawn to scale what is the measure of the minor arc bc? a 35° b 110° c 145° d 180°

Explanation:

Step1: Recall circle - arc relationship

The measure of an arc is equal to the measure of its central - angle. The central angle of arc \(BC\) can be found using the fact that the sum of angles around a point is \(360^{\circ}\) and \(AC\) is a diameter (\(\angle APC = 180^{\circ}\)).

Step2: Calculate the central angle of arc \(BC\)

We know that \(\angle APB=35^{\circ}\), and \(\angle APC = 180^{\circ}\). So, \(\angle BPC=\angle APC-\angle APB\). Substituting the values, we get \(\angle BPC = 180 - 35=145^{\circ}\). Since the measure of the minor arc \(BC\) is equal to the measure of the central angle \(\angle BPC\), the measure of the minor arc \(BC\) is \(145^{\circ}\).

Answer:

C. \(145^{\circ}\)