Pencil, highlighter, red pen, GP NB, calculator U10D4 Have out: Bellwork: Find the following measures for circle F. A C B F E 24° a) m EA m EA + m AB + m ECB = 360° m EA + 24° + 208° = 360° (circle = 360°) m EA + 232° = 360° m EA = 128° +1 b) m BC m BC + m EA + m AB = 180° (semicircle = 180°) m BC + 128° + 24° = 180° m BC + 152° = 180° m BC = 28° +1 208° c) d) (Inscribed  Thm) +1 +1 (arc measure = central ) +1 +1

Add to your notes... Tangent and Secant Secant: A line that intersects a circle in two points Tangent: A line that intersects a circle at only one point

b) What appears to be true about the centers of all the circles? CS – 30 a) Draw a figure like the one below in your notebook (i.e., circle C tangent to line L at D). Draw 3 more circles of different sizes tangent to the line L at the same point D (that is, one line, one point D, four different size circles). Draw the radius of each circle to the point of tangency, D. b) What appears to be true about the centers of all the circles? They are on the same line, which appears to be vertical. D L C

The radius appears to be perpendicular to the tangent. CS – 30 c) Draw the radius of the largest circle from its center to point D. What appears to be true about the angle between the radius and the tangent? The radius appears to be perpendicular to the tangent. d) Write a conjecture stating the relationship between a radius of a circle drawn to a point of tangency and the tangent line. Any tangent of a circle is perpendicular to the radius of the circle at the point of contact, i.e. tangency. D L C

Tangent / Radius Theorem Add to your notes... Tangent / Radius Theorem Any tangent of a circle is perpendicular to a radius of the circle at the point of tangency. >>

Solve the following problems. CS – 32 Solve the following problems. a) If AT is tangent to circle C, find AC and the area of the circle. b) If PI is tangent to circle G, find PG and the area of the circle. C A 5 4 T G 3 P I (Tangent / Radius Thm) (def. of ) (Tangent / Radius Thm) mP=90° mA=90° (def. of ) (Pythagorean Thm) A = r2 (Pythagorean Thm) (AC)2 + 42 = 52 A = r2 (AC)2 + 16 = 25 A = 3 u2 A =  • 32 (AC)2 = 9 A = 9 u2 A  9.42 u2 AC = 3 u A  28.27 u2

2) PA is tangent to circle R at point E 2) given CS – 33 In the figure, PA is tangent to circle R at E and PE = EA. Is PER  AER? Prove it or show why not. P E A Statement Reason R 1) 1) given 2) PA is tangent to circle R at point E 2) given 3) 3) Tangent / Radius Thm 4) 4) Def. of perpendicular 5) 5) reflexive property 6) PER  AER 6) SAS

Use the figure at right to find m DAB and m DCB. CS – 34 Use the figure at right to find m DAB and m DCB. 53° A D C B (Inscribed Angle Thm) (circle = 360°) a) Use your results to find mA. (Inscribed Angle Thm) b) Fred has a shortcut because he notices a relationship between the measures of angles A and C. What is the relationship? mA + mC = 127° + 53° = 180° They are supplementary.

c) Do you think B and D will have the same relationship? Prove it! 53° A D C B CS – 34 c) Do you think B and D will have the same relationship? Prove it! 127 Yes Statement Reason 1) 1) Sum of int. s of n- gon = 180º(n – 2) 2) 2) Substitution 3) 3) Subtraction 4) B & D are supplementary 4) Def of supplementary d) Prove that the opposite angles of any quadrilateral inscribed in a circle must be supplementary. Together the two angles intercept the entire circle, so their sum must be ½(360°) = 180° by the Inscribed Angle Theorem.  The opposite angles in a quadrilateral inscribed in a circle are supplementary.

Work on CS 37 & the worksheet.

The tangent lines appear to be perpendicular to the diameter. CS – 31 Draw a circle and any diameter. At each endpoint of the diameter, draw a tangent line. a) What appears to be the relationship between a diameter and tangent lines that intersect at the end(s) of a diameter? The tangent lines appear to be perpendicular to the diameter. b) What appears to be the relationship between the pair of tangent lines, one at each end of a diameter? The tangent lines appear to be parallel to one another.

1) Tangent lines  to diameter. 1) given CS – 31 c) Prove that the tangents drawn at the endpoints of a diameter are parallel, given that they are  to the diameter. Statement Reason 1) Tangent lines  to diameter. 1) given 2) All s formed by tangents & diameter = 90º 2) def. of  3) Alternate Interior s = then tangent lines //. 3) Converse of the Alternate Interior  Thm