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Difficulty: MediumCircle Geometry

Points AA, BB, and CC lie on a circle with center OO. The length of the minor arc ACAC is 49\frac{4}{9} of the circumference of the circle. What is the measure, in degrees, of the inscribed angle ∠ABC\angle ABC?

Answer: 80 degrees

Answer

The measure of the inscribed angle is 80 degrees.
The correct answer is 80. The minor arc ACAC constitutes 49\frac{4}{9} of the circle's circumference, which corresponds to an arc measure of 49×360∘=160∘\frac{4}{9} \times 360^\circ = 160^\circ. By the Inscribed Angle Theorem, the measure of the inscribed angle ∠ABC\angle ABC is half the measure of the intercepted arc, which is 12×160∘=80∘\frac{1}{2} \times 160^\circ = 80^\circ.

Step-by-Step Solution

1
Determine the degree measure of the minor arc ACAC.
The measure of minor arc ACAC is 160∘160^\circ.
Since a full circle has a circumference corresponding to 360∘360^\circ, minor arc ACAC has a degree measure of 49×360∘=160∘\frac{4}{9} \times 360^\circ = 160^\circ.
2
Calculate the measure of the inscribed angle ∠ABC\angle ABC.
The measure of ∠ABC\angle ABC is 80∘80^\circ.
According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of the arc it intercepts. Thus, the measure of ∠ABC\angle ABC is 12×160∘=80∘\frac{1}{2} \times 160^\circ = 80^\circ.

Key Concept

Inscribed Angle Theorem and Arc Measure
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Topics

Circle Geometry

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