Tangent and Secant Lines

Tangent to Circles - Problem Solving


As shown in the above diagram, triangle ABCABC inscribed in a circle is an isosceles triangle with AB=AC,\lvert{\overline{AB}}\rvert=\lvert{\overline{AC}}\rvert, where AB\lvert{\overline{AB}}\rvert denotes the length of AB.{\overline{AB}}. Given the following two lengths: AP=11 and PQ=5,\lvert{\overline{AP}}\rvert=11 \text{ and } \lvert{\overline{PQ}}\rvert=5, what is AB2?{\lvert\overline{AB}\rvert}^2?

Note: The above diagram is not drawn to scale.

The above diagram illustrates a 1717 meters by 1010 meters rectangular farmland ABCD.ABCD. The farmland contains a circular well OO with radius 44 meters at its upper left corner, which is tangent to AB\overline{AB} and AD.\overline{AD}. If the farmer wants to build a straight fence dividing the land into two areas in the following way, what should be the length of FD\overline{FD} (in meters):

  • FEC=45\angle FEC = 45 ^\circ
  • the well goes to the left of the fence
  • make the area of FECDFECD as large as possible?

ABCD\square ABCD in the above diagram is a square-shaped park with side length 30230\sqrt{2} meters. If there is a circular fountain with radius 1515 meters at the center of the park, what is the length (in meters) of the shortest path from AA to CC that does not pass through the fountain?

DE\overline{DE} is tangent to circle OO at point C,C, and points AA and BB lie on circle O.O. If ACD=79\angle ACD=79 ^\circ and OBA=2OAC,\angle OBA=2\angle OAC, what is the measure (in degrees) of OCB?\angle OCB?

You have a piece of rectangular land ABCDABCD that measures 3030 meters by 2525 meters. There are two wells, one at the upper left corner and another at the lower right corner, with the same radius. Both wells are tangent to the borders of the land. When you put a fence EFEF that is tangent to both wells, as shown in the above diagram, you find EFB=60.\angle EFB=60^\circ. What is the radius of the two wells?


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