If AB is a diameter of a circle and C is any point on the circle, what is the measure of angle ACB?

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Prepare for the AMSOC 26-003 Module A Test. Utilize flashcards and multiple choice questions with hints and explanations. Ace your exam!

Multiple Choice

If AB is a diameter of a circle and C is any point on the circle, what is the measure of angle ACB?

Explanation:
If a line through the circle’s center connects two points, that line is a diameter, and any angle formed by joining those endpoints to a third point on the circle is a right angle. This is Thales’ theorem: an angle subtending a diameter is 90 degrees. Reason: the central angle subtending the same arc AB is 180 degrees (OA and OB are opposite radii). An inscribed angle like angle ACB intercepts arc AB, so its measure is half of the central angle that intercepts the same arc. Half of 180 degrees is 90 degrees. This holds for any position of C on the circle, as long as AB remains the diameter. Therefore angle ACB measures 90 degrees.

If a line through the circle’s center connects two points, that line is a diameter, and any angle formed by joining those endpoints to a third point on the circle is a right angle. This is Thales’ theorem: an angle subtending a diameter is 90 degrees.

Reason: the central angle subtending the same arc AB is 180 degrees (OA and OB are opposite radii). An inscribed angle like angle ACB intercepts arc AB, so its measure is half of the central angle that intercepts the same arc. Half of 180 degrees is 90 degrees. This holds for any position of C on the circle, as long as AB remains the diameter. Therefore angle ACB measures 90 degrees.

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