Geometry

Tangent and Secant Lines

Tangent - Perpendicular to Radius

         

What is the area of the shaded quadrilateral shown above?

Two circles MM and NN are tangent to two perpendicular lines that intersect at point L,L, as shown in the above diagram. If the radii of the two circles are r1=16r_{1}=16 and r2=8,r_{2}=8, respectively, what is the distance between MM and N?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 2424 and the radius of the base circle of the cone is 48,48, what is the height of the cone?

Note: the above diagram is not drawn to the scale

AN\overline{AN} and AB\overline{AB} are both tangent to circle O.O. If the radius of the circle is 4444 and NAB=30,\angle NAB={30}^\circ, what is the distance between NN and AB?\overline {AB}?

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

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

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