Tangent and Secant Lines

Tangent and Secant Lines: Level 3 Challenges


O O is the center of the circle. If AB=18 AB = 18 cm, then the area of the brown part is xπ x \pi . What is xx?

ABCDABCD is a cyclic quadrilateral with AB=11\displaystyle \overline{AB}=11 and CD=19\displaystyle \overline{CD}=19. PP and QQ are points on AB\overline{AB} and CD \overline{CD}, respectively, such that AP=6\displaystyle \overline{AP}=6, DQ=7\displaystyle \overline{DQ}=7, and PQ=27.\displaystyle \overline{PQ}=27. Determine the length of the line segment formed when PQ\displaystyle \overline{PQ} is extended from both sides until it reaches the circle.

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You are currently located on point (0,0)(0,0) and you want to get on point (54,18)(54,18). However, there are some annoying circular objects in the way! They are defined by x2+y218x18y+81=0x^2+y^2-18x-18y+81=0 x2+y290x18y+2025=0x^2+y^2-90x-18y+2025=0

If you cannot walk through these annoying circular objects, then the shortest possible path possible to point (54,18)(54,18) can be expressed as a+bc+dπa+b\sqrt{c}+d\pi for positive integers a,b,c,da,b,c,d with cc square-free. What is a+b+c+da+b+c+d?

Circles of radii 3,4, and 5 units are externally tangent. The lines which form the 3 common external tangent intersect at P,P, which is equidistant from the 3 points of tangency. Find this distance (from PP to any point of tangency)?

In ABC,\triangle ABC, AB=6AB=6 , BC=4BC = 4 and AC=8. AC=8. A segment parallel to BC \overline{BC} and tangent to the incircle of ABC\triangle ABC intersects AB\overline{AB} at MM and AC\overline{AC} at NN.

If MN=abMN=\dfrac{a}{b}, where aa and bb are co-prime positive integers, what is the value of a+ba+b ?


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