Geometry

Inscribed and Circumscribed Figures

Incircle of Triangle

         

In the above diagram, ABC\triangle ABC is an equilateral triangle and point OO is the center of the circle inscribed in ABC.\triangle ABC. If the length of AB\overline{AB} is AB=27,\lvert \overline{AB} \rvert = 27, AB and OD\overline{AB} \text{ and } \overline{OD} are parallel, and AC and OE\overline{AC} \text{ and } \overline{OE} are parallel, what is DE(=x)?\lvert\overline{DE}\rvert(=x)?

In the above diagram, circle OO is the incircle of right triangle ABC.ABC. Given the two side lengths AB=7 and BC=16,\lvert\overline{AB}\rvert = 7 \text{ and }\lvert\overline{BC}\rvert = 16, what is the area of the circle?

Note: The above diagram is not drawn to scale.

In the above diagram, circle OO is inscribed in triangle ABC,\triangle ABC, with points of contact D,E,D, E, and FF. If DBE=54 and DAF=46,\angle DBE = 54^{\circ} \text{ and } \angle DAF = 46^{\circ}, what is BOC?\angle BOC?

Note: The above diagram is not drawn to scale.

In the above diagram, circle OO is inscribed in ABC\triangle ABC, with points of contact at points D,ED, E and FF. If a=4,b=7a=4, b=7 and c=9,c=9, what is the perimeter of triangle ABC?\triangle ABC?

Note: The above diagram is not drawn to scale.

In the above diagram, circle OO with radius r=20r=20 is inscribed in ABC\triangle ABC. If the perimeter of ABC\triangle ABC is xx and the area of ABC\triangle ABC is yy, what is yx?\frac{y}{x}?

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