Question

Line segment BD is a diameter of circle E.

Circle E is inscribed with triangle B C D. LIne segment B D is a diameter. Line segments D C and C B are secants. Angle D B C is 51 degrees.
What is the measure of arc B C?

39°
78°
102°
129°

Answer 1

78 degrees

Explanation:

Mainly because the measure of the arcs intercepted by the interior angles equals 2 times those interior angles.

so arc DC = 2 * angle DBC = 2 * 51 degrees = 102 degrees.

We also see that because BD is a diameter the angle BCD that intercepts it is 90 degrees because half-circle arc BD = 2 * angle BCD

180 degrees = 2 * angle BCD

also we want arc BC.

we know that the total circle is the sum of all 3 arcs.

arc BC + (half-circle) arc BD + arc DC = 360 degrees

arc BC + 180 + 102 = 360 degrees.

arc BC = 360 - 180 - 102 = 78 degrees