Geometry

Inscribed and Circumscribed Figures

Circumcircle of Triangle

         

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

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

Note: The above diagram is not drawn to scale.

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

Note: The above diagram is not drawn to scale.

In the above diagram, point OO is the circumcenter of ABC\triangle ABC and ABC\angle ABC is 15.15^{\circ}. If the radius of the circumscribed circle is 16,16, what is the area of ABC?\triangle ABC?

Note: The above diagram is not drawn to scale.

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

Note: The above diagram is not drawn to scale.

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