Circles SAT question

Math · Geometry and Trigonometry · Easy · Answer it here, explanation included.

The figure presents a circle with center O. Points A, and C are indicated on the circle, creating minor arc A, C. A diameter is drawn from point A to a point on the other side of the circle. Similarly, a diameter is drawn from point C to a point on the other side of the circle. The two lines intersect at the origin, forming a right angle at angle A, O C

The circle above with center O has a circumference of 36. What is the length of minor arc $A, C$ ?

ExplanationShow

Choice A is correct. A circle has 360 degrees of arc. In the circle shown, O is the center of the circle and $angle A, O C$ is a central angle of the circle. From the figure, the two diameters that meet to form $angle A, O C$ are perpendicular, so the measure of $angle A, O C$ is $90 degrees$ . Therefore, the length of minor arc $A, C$ is $the fraction 90 over 360$ of the circumference of the circle. Since the circumference of the circle is 36, the length of minor arc $A, C$ is $the fraction 90 over 360, end fraction, times 36, equals 9$ .

Choices B, C, and D are incorrect. The perpendicular diameters divide the circumference of the circle into four equal arcs; therefore, minor arc $A, C$ is $one fourth$ of the circumference. However, the lengths in choices B and C are, respectively, $one third$ and $one half$ the circumference of the circle, and the length in choice D is the length of the entire circumference. None of these lengths is $one fourth$ the circumference.

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Circles SAT question

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