Radius, diameter, circumference, π (Pi), central angles, Pythagorean relationship. about CIRCLES

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1 Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 1 Chapter 8 in textbook (p ) 5/50 or 10% on 2011 CRT: 5 Multiple Choice WHAT YOU SHOULD ALREADY KNOW: Radius, diameter, circumference, π (Pi), central angles, Pythagorean relationship THE 3 BIG IDEAS: about CIRCLES (1) Tangents to circles (2) Chord properties in circles (3) Inscribed & central angle relationships in circles THE 4 MAIN CIRCLE PROPERTIES: (1) A tangent to a circle is perpendicular to the radius at the point of tangency. (2) The perpendicular from the center of a circle to a chord bisects the circle (3) The measure of the central angle is equal to TWICE the measure of the inscribed angle subtended by the same arc. (4) The inscribed angles subtended by the same arc are congruent. *** *** You must solve problems AND justify (explain) HOW you get your solution using these 4 Circle properties ***

2 Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 2 4 main CIRCLE PROPERTIES : (1) A tangent to a circle is perpendicular to the radius at the point of tangency. [SECTION 8.1 p ] Q: What does tangent mean? A: A straight line that just touches the curve or circle. Q: What does perpendicular mean? A: When line segments intersect to form 90 0 RIGHT angles Q: What is a radius? r = d 2 A: ANY line from the center of the circle to anywhere on the outside of the circle. [ There are infinite numbers of radius called radii for plural in a circle] Q: What is a diameter? D = 2 x r A: The distance from 1 point on outside of a circle to the other side BUT it MUST pass through the CENTER of the circle Q: What is the Point of Tangency? A: The point at which the tangent line JUST touches the circumference of the circle.

3 Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 3 CIRCLE PROPERTY # 1 continued... A tangent to a circle is perpendicular to the radius at the point of tangency. {Section 8.1 p. 384: Properties of tangents to a circle} INVESTIGATION p. 384 CONNECT p. 385 NOTES: Q: What is the relationship between the tangent of a circle and the radius at the point of tangency A: IMPORTANT RESULT: If you can show that there is a 90 0 right angle inside a triangle THEN can use PYTHAGOREAN RELATIONSHIP to find missing side lengths***

4 Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 4 CIRCLE PROPERTY # 1 continued... A tangent to a circle is perpendicular to the radius at the point of tangency. {Section 8.1 p. 384: Properties of tangents to a circle} PRACTICE TEXTBOOK QUESTIONS: P : # 3,4 Finding the measure of an angle in a triangle: {Ex 1 p. 386, p. 388 # 5,7, Using the Pythagorean Relationship in a circle : {Ex 2 p. 386, p. 388 # 6,8,9, Solving problems using the Tangent and radius property above along with the Pythagorean relationship: {Ex 3 p. 387, p. 389 # 12,13,14,16c, 17} [ATTACH THESE QUESTIONS after this page please!!!] SEE MID UNIT REVIEW p. 403

5 Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 5 CIRCLE PROPERTY # 2: [SECTION 8.2 p ] (2) The perpendicular from the center of a circle to a chord bisects the circle Q: What does perpendicular mean? A: SEE p. 2 Q: What is the center of a circle? A: The center of a circle is in the middle equidistant from any point on the circumference Q: What is a chord? A: Any straight line segment from one side of a circle to the other Q: What does bisect mean? A: Bisect means to cut into 2 pieces ( bi meaning 2)

6 Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 6 CIRCLE PROPERTY # 2 continued... The perpendicular from the center of a circle to a chord bisects the chord INVESTIGATION p. 392 CONNECT p. 393 Q: What is the relationship between the perpendicular line from the center of the circle and a chord? A: 3 IMPORTANT RESULTs: [p. 393 YELLOW BOX!!!] ***************** Perpendicular to chord property 1 [p. 393 yellow box] Perpendicular line from center BISECTS the chord into 2 equal parts Perpendicular to chord property 2 [p. 393 yellow box] Perpendicular Bisector of a chord passes through the center of the circle Perpendicular to chord property 3 [p. 394 yellow box] A line that joins the center of the circle to the midpoint of a chord is perpendicular to the chord

7 Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 7 CIRCLE PROPERTY # 2 continued... The perpendicular from the center of a circle to a chord bisects the chord TEACHER GUIDE questions: 1. Explain how to find the center of a circle, given any 2 chords in a circle that are NOT parallel? 2. Investigation p. 392 Folding activity to develop this circle property # 2 PRACTICE TEXTBOOK QUESTIONS:{ NEED LOTS OF EXTRA PRACTICE outside of the textbook!!!} P. 397 # 3, Finding missing angles in a triangle in a CIRCLE p. 394 Ex 1, p. 397 #4, p. 403 # 1, 4, Using the Pythagorean Relationship in a triangle in a CIRCLE p. 395 Ex 2, p. 397 # 5, 6, 7, p. 403 # 2,3,5,6,7 Solving problems using Circle Property # 2 (The perpendicular from the center of p. 395 Ex 3, a circle to a chord bisects the chord)

8 Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 8 CIRCLE PROPERTY # 3 & 4 together [SECTION 8.3 p ] (3) The measure of the central angle is equal to TWICE the measure of the inscribed angle subtended by the same arc. (4) The inscribed angles subtended by the same arc are congruent. Q: What is a central angle? A: A central angle is an angle with endpoints and located on a circle's circumference and vertex O located at the circle's center Q: What is an inscribed angle? A: An inscribed angle is an angle formed by points A, B, and C on the circle's circumference. Q: What is an arc? A: An arc of a Circle is a curved line drawn between two points along the circumference of a Circle.

9 Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 9 CIRCLE PROPERTY # 3 & 4 together continued... The measure of the central angle is equal to TWICE the measure of the inscribed angle subtended by the same arc. The inscribed angles subtended by the same arc are congruent. *** Any angle that subtends an arc is an angle that is formed by the endpoints of the arc Q: What does subtended by the same arc mean? A: (i) If 2 or more INSCRIBED angles are subtended (share) by the same arc they are EQUAL! (ii) If one angle is INSCRIBED and the other angle is a CENTRAL angle then: The measure of the central angle is equal to TWICE the measure of the inscribed angle subtended by the same arc. Angle AOC = Angle ABC OR Angle ABC = Angle AOC *****

10 Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 10 CIRCLE PROPERTY # 3 & 4 together continued... The measure of the central angle is equal to TWICE the measure of the inscribed angle subtended by the same arc. The inscribed angles subtended by the same arc are congruent. NOTES: NOTES: NOTES:

11 Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 11 CIRCLE PROPERTY # 3 & 4 together continued... The measure of the central angle is equal to TWICE the measure of the inscribed angle subtended by the same arc. The inscribed angles subtended by the same arc are congruent. Inscribed angle subtended by a diameter = 90 0 TEACHER GUIDE questions: 1. After a power outage, Mackenzie helps her father by shining a flashlight beam on his work. Her flashlight projects light through an angle of 15 0, while her father s flashlight projects light through an angle of Explain, using a diagram, where Mackenzie would stand so that her flashlight would light up the SAME area as her father s flashlight.

12 Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 12 CIRCLE PROPERTY # 3 & 4 together continued... The measure of the central angle is equal to TWICE the measure of the inscribed angle subtended by the same arc. The inscribed angles subtended by the same arc are congruent. PRACTICE TEXTBOOK QUESTIONS:{ NEED LOTS OF EXTRA PRACTICE outside of the textbook!!!} 3 YELLOW BOXES p angle properties Given either the inscribed angle OR the central angle, Find the other angle p. 407 Ex 1, p. 410 # 3,4, Finding angles in an inscribed triangle in a circle p. 408 Ex 3, p. 410 # 5,6,11

13 Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 13 CIRCLE PROPERTY # 3 & 4 together continued... TEACHER GUIDE questions: 1. If measure of < ABC = find the measure of < AB C = because 2. Given that < ABD = 56 0, what are the measures of x (angle ACD) and y (angle AED)? < x = because < y = because B Cc E A D

14 Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 14 CIRCLE PROPERTY # 3 & 4 together continued... TEACHER GUIDE questions continued Given < ABE = 6x 14 0 AND < EDA = 4x Find the missing measures (a) What is the value of x? (b) What is the measure of < ABE = because (c) What is the measure of < ADE = because (d) What do you notice about these 2 angles? Explain. B D C A A A E

15 Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 15 PRACTICE QUESTIONS: DO THESE QUESTIONS PLEASE!!! (1) Study Guide p (2) Unit 8 REVIEW p # 1,2,3,5,6,8,9,10 (3) EXTRA PRACTICE WORKSHEETS with answer keys *** (4) ANY EXTRA questions given or assigned in class NOTES: