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# Inscribed and Circumscribed Figures

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

# Incircle of Triangle

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

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

Note: The above diagram is not drawn to scale.

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

Note: The above diagram is not drawn to scale.

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

Note: The above diagram is not drawn to scale.

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

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