Geometry

# Tangent and Secant Lines: Level 3 Challenges

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

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

Note: The image is not drawn to scale.

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

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

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

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

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

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