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Inscribed and Circumscribed Figures

Learn about packing shapes neatly into others, and the resulting geometric properties.

Circumcircle of Triangle

In the above diagram, point \(O\) is the circumcenter of \(\triangle ABC\) and the lengths of \(\overline{AB}\) and \(\overline{BC}\) are the same, i.e. \(\lvert \overline{AB} \rvert=\lvert \overline{BC} \rvert.\) If \(\angle OAB(=a)\) is \(15^{\circ},\) what is \(\angle OAC(=x)\)?

In the above diagram, point \(O\) is the circumcenter of \(\triangle ABC\) and point \(O'\) is the circumcenter of \(\triangle AOC.\) If \(\angle ABO = 28^{\circ},\) what is the value of \(\angle OO'C\) in degrees?

Note: The above diagram is not drawn to scale.

In the above diagram, point \(O\) is the circumcenter of \(\triangle ABC.\) The length of \(\overline{AF}\) is \(10(=a)\) and the length of \(\overline{OF}\) is \(3(=b).\) If the area of \(\triangle ABC\) is \(116,\) what is the area of \( CDOE?\)

Note: The above diagram is not drawn to scale.

In the above diagram, point \(O\) is the circumcenter of \(\triangle ABC\) and \(\angle ABC\) is \(15^{\circ}.\) If the radius of the circumscribed circle is \(16,\) what is the area of \(\triangle ABC?\)

Note: The above diagram is not drawn to scale.

In the above diagram, point \(O\) is the circumcenter of \(\triangle ABC.\) The length of \(\overline{AF}\) is \(6(=a)\) and the length of \(\overline{OF}\) is \(5(=b).\) If the area of \(\triangle ABC\) is \(98,\) what is the area of \( CDOE?\)

Note: The above diagram is not drawn to scale.

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