Question

line segment bd is a diameter of circle e. what is the measure of arc bc? 39° 78° 102° 129°

Explanation:

Step1: Recall the inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. In circle \(E\), \(\angle BDC = 51^{\circ}\) is an inscribed angle and arc \(BC\) is the intercepted arc.

Step2: Use the formula for the relationship between inscribed - angle and arc

Let the measure of arc \(BC\) be \(x\). According to the inscribed - angle theorem, \(\angle BDC=\frac{1}{2}\text{measure of arc }BC\). So \(x = 2\angle BDC\).

Step3: Calculate the measure of arc \(BC\)

Substitute \(\angle BDC = 51^{\circ}\) into the formula. We get \(x=2\times51^{\circ}=102^{\circ}\).

Answer:

C. \(102^{\circ}\)