Geometry Toolkit for Circle Theorems Exploration

1. HW, red pen, ruler, pencil, highlighter, calculator, GP NB r U10D5 Have Out: Bellwork: Complete the bellwork in your packet. Here are the justifications we have learned in this unit so far: • Circle = 360º • Semicircle = 180 º • Inscribed  Thm • Opposite s of a quadrilateral inscribed in a circle are supplementary. • Tangent/Radius Thm • Central  = arc measure • Inscribed  of semicircle = 90º • Arc Length = • Area of a sector =

2. m ML = 2 (mMNL) m ML = 2(79°) +1 +1 m MOL = m ML m AOB = m AB 3. 1. 2. P N 58 A 79° x O R M 52° B x O x C Q L (Inscribed  Thm) (Inscribed  Thm) 1. 2. 3. (arc measure = central ) = 158° x = 52° (arc measure = central )

3. +1 +1 +1 +1 +1 D E G S x 114° 60° T x H 16 V O O x U F 6. 5. 4. 114 x 60 x 4. (Tangent/Radius Thm) (Opposite s of quadrilateral inscribed in a circle are supplementary ) (def. of ) mEDO = 90 30 - 60 - 90 SL: n = 8u 6. x + 114 = 180° LL: x = 66° Hyp: 2n = 16u (Inscribed  of semicircle = 90°) x = 8u 5.

4. Add to your Vocab Toolkit: Chord: Line segment with its endpoints on the circle. For the next activity, you will need your ruler, a pencil, and the resource page in your packet.

5. Draw a chord on your circle (not near the center). CS 40 • Fold the circle in ½ so that the ends of the chord touch each other. Crease the fold & then unfold your circle. a) How does the folded line appear to relate to the circle and the chord? What appears to be true about the parts of the chord? About the arcs? Write a conjecture for these questions. It is a diameter because it splits the circle in half. It appears to bisect the chord perpendicularly. It also appears to bisect the arcs.

6. CS 40 b) Use your ruler to measure the two pieces of chord. Write the measure of each piece of the chord on your circle. Draw another chord (not parallel to the first one). • Fold the circle in ½ so that the ends of the chord touch each other. c) What appears to be true about the spot where the folded lines intersect? Since the folded lines are both diameters, they meet at the center of the circle. • Draw another chord and do the folding again. Do all three folding lines intersect at the center of the circle? YES! By definition, the diameters must go through the center of the circle.

7. In K, is bisected. Statement Reason Prove that 9) CS 41 R W A E 1) is bisected 1) given 2) 2) Def. of bisect K 3) 3) reflexive property 4) radii of circle are  N 4) 5) AWK  EWK 5) SSS 6) AWK  EWK 6) s  parts. 7) Linear Pair 7) mAWK+mEWK =180º 8) mAWK =mEWK = 90º 8) Substitution & Division 9) Definition of 

8. In B, . Statement Reason 6) 7)bisects Prove that bisects . CS 42 R 1) 1) given I N 2) =90° 2) Def. of perpendicular B H 3) 3) radii of circle are  A 4) 4) reflexive property 5) BRH  BAH 5) HL 6)  s  parts 7) Def. of bisect

9. Add to your notes: Diameter/Chord Theorems • If a diameter bisects a chord, then it is perpendicular to the chord. • If a diameter is perpendicular to a chord, then it bisects the chord. Is this theorem valid if the term radius replaces the term diameter? CS 43 Yes, the Diameter/Chord Thm still applies for a radius if the radius cuts the chord!

10. In P, . If PU = 4 and m PNU = 30°, find the lengths of the sides below. Show all steps leading to your solution. Short Leg: n Long Leg: n Hyp: 2n CS 44 mRUP = mNUP=90° (Def. of perpendicular) E N a) UN b) UR 30° U 30- 60- 90 Triangle: UN = UR 4 (Diameter/ Chord Thm) P R = PN c) PE d) UE = 4 u = 8 u PE = PN UE + PU = PE UE = PE – PU (All radii are equal) = 8 – 4

11. Work on assignment... CS 45, 48 (graph only), 49, 61, Vocab WS, & MCFR #28-32