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Geometry

Tangent and Secant Lines

Challenge Quizzes

Two circles $$M$$ and $$N$$ are tangent to two perpendicular lines that intersect at point $$L,$$ as shown in the above diagram. If the radii of the two circles are $$r_{1}=16$$ and $$r_{2}=8,$$ respectively, what is the distance between $$M$$ and $$N?$$

Note: The above diagram is not drawn to scale.

In the above diagram, a sphere is inscribed in a right circular cone. If the radius of the sphere is $$24$$ and the radius of the base circle of the cone is $$48,$$ what is the height of the cone?

Note: the above diagram is not drawn to the scale

$$\overline{AN}$$ and $$\overline{AB}$$ are both tangent to circle $$O.$$ If the radius of the circle is $$44$$ and $$\angle NAB={30}^\circ,$$ what is the distance between $$N$$ and $$\overline {AB}?$$

In the above diagram, $$\triangle ABC$$ and $$\triangle DEF$$ are both equilateral triangles. If the circle in the diagram with center $$O$$ is the circumscribed circle of $$\triangle DEF$$ and the inscribed circle in $$\triangle ABC,$$ what is the ratio $$\lvert\overline {AB}\rvert : \lvert\overline {DE}\rvert?$$

Note: $$\lvert\overline {AB}\rvert$$ denotes the length of $$\overline {AB}.$$

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