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Properties of Circles

What is pi? If the area of a circle is 49π, what is its circumference? If a circle is inscribed in a square, what is the ratio of the areas of the circle and the square?

Inscribed Angle

         

\(A, B, C\) and \(D\) are consecutive points on a circle (labeled clockwise) such that \(\angle ABC = 73 ^\circ\). What is the measure of \(\angle ADC\) (in degrees)?

The above diagram shows a circle of radius \(r,\) with \(\angle ABC=15^\circ.\) If the length of arc \(\widehat{AC}\) is equal to \(10,\) what is the measure of \(r?\)

Note: The above figure is not drawn to scale.

In the above diagram, point \(O\) is the center of the circle. If \(\angle BDC=26^\circ,\) what is \(\angle BCA\) in degrees?

Note: The above figure is not drawn to scale.

In the diagram above, the measures of the two arcs \(\widehat{BC}\) and \(\widehat{CD}\) are equal. If \(\angle BAC=20^\circ\) and \(\angle ABD=40^\circ,\) and the measure of arc \(\widehat{BC}\) is equal to \(6,\) what is the measure of arc \(\widehat{APB}?\)

Note: The above figure is not drawn to scale.

In the diagram above, lines \(AB\) and \(BC\) are tangent lines of a circle, with points \(A\) and \(C\) as their respective tangent points. If \(\angle ADC=70^\circ,\) what is \(\angle ABC\) in degrees?

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