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Given a circle in which the diameter AB equals 4cm. If two points C and D lie on the circle and the angle ABC= 18 degrees
and angle BAD=36 degrees, find the length of the major arc CD.
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The answer and the situation depend on whether the points C and D lie in one side of the diameter AB or in different sides of AB.
1. Let's assume that the points A and B lie in one side of the diameter AB.
Let O be the center of the given circle.
Since the measure of an inscribed angle in a circle is half the measure of the corresponding central angle, we have
the measure of the central angle AOC is equal to 2*18° = 36°, and
the measure of the central angle BOD is equal to 2*36° = 72°.
Thus the measure of the central angle COD is 180° - 36° - 72° = 72°, and
the measure of the major arc CD is 72°.
2. If the points A and B lie in different sides of the diameter AB then
the measure of the major arc CD is 180° + 36° - 72° = 144°.
Answer. Two answers: 72° and/or 144°.