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Geometry

Inscribed and Circumscribed Figures

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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