Name __________________ Date __________________ Honors Geometry Unit 6 Test Circles 1. SHORT ANSWER Find the indicated lengths and angles. Show calculations because partial credit will be awarded for reasonable progress, but no detailed annotations or proofs are necessary. Note: figures not necessarily drawn to scale. a. Given: FD 9 , ED 3 , CB 5 , BA 10 , BA is a tangent Find: CD and DA E F D C B A b. Given: ABCD is an inscribed quadrilateral, with BC CD and BC AD . PQ is tangent to the circle at A. mBAQ 32 . Find: mBAD and mBCD c. Given: B , with radius 5 and the tangent segment CD 12 , and DE FE . Find: Length of DE and the LENGTH of FE (the arc length). d. A half-circle is inscribed within an equilateral triangle such that the diameter of the halfcircle is centered on one edge of the triangle and the arc is tangent to the other two sides. What is the diameter of the semi-circle if the triangle has an edge length of 4? 2. SHORT JUSTIFICATION You do not need to write formal proofs, but your explanations should make it clear that you understand the underlying geometric ideas. Brief explanations and/or annotated calculations are sufficient. a. PB is tangent to the circle at point B. Briefly explain how you can find the measure of angle x WITHOUT using the secant-tangent angle theorem. B 160 35 x P C A b. The steps and diagrams below show you how to construct a tangent to a circle from an arbitrary point outside the circle. Briefly explain why the segment drawn in the final step is guaranteed to be tangent to the original circle. Given C and point P outside the circle. 1. Construct segment PC. 2. Construct M, the midpoint of PC. 3. Construct M, centered at M with radius CM. 4. Place point I at one intersection of C and M. 5. Construct IP. P C P C M I C P M c. Circle C has a radius of R and x and y denote the lengths of the two parts of the chord. Briefly explain why x y R 2 d 2 . Hint – you probably need to add some segments to the diagram. Do not spend too much time on this one. It is challenging to see, so it will be weighted less. x y d C R 3. PROOF (JUSTIFICATIONS) F C In the diagram to the right, BF and BE are tangents to A and AC and AD are tangents to B . The “eyeball theorem” states that given the information above, MN PQ . M P B A Q N D While the entire proof is too long to do here, we’ll have you do pieces of it below. E a. Given the information above, prove that mAFB mACB 90 and FAC FBC (call this angle 2 ). F C M P B A N Q D E b. Using what you proved in part a (or just assuming a to be true), prove mMFP mMCP . F C M P B A N Q D E c. It can be shown that FCPM is an inscribed quadrilateral. Use this (and the results in a and b if needed) to prove that mCMP mCAB . Call this angle . F C M P B A N Q D E d. It can also be shown that ABCF is an inscribed quadrilateral. Use this (along with the results in a, b, or c if needed) to prove that BAC BFC . F C M P B A N Q D E Note that from here utilizing corresponding angles, CMP and CAB , it can be shown that MP // AB // NQ . Moreover we can show that MN AB and PQ AB , and therefore MPQN is a rectangle and consequently MN PQ as we originally set out to prove. We’ll leave the details to you as a fun, optional homework challenge should you choose to tackle it.