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Chapter 9 Circles • Define a circle and a sphere. • Apply the theorems that relate tangents, chords and radii. • Define and apply the properties of central angles and arcs. 9.1 Basic Terms Objectives • Define and apply the terms that describe a circle. The Circle is a set of points in a plane at a given distance from a given point in that plane B A Blackout slide to show points The Circle The given distance from the center to any point on the circle is a radius (plural radii) How many radii does a circle have? Are they all the same length? B A center A “Circle with center A” Think – Pair - Share •What is the difference between a line and a line segment? Chord any segment whose endpoints are on the circle. C B A Diameter A chord that contains the center of the circle B A C What is another name for half of the diameter? Secant any line that contains a chord of a circle. C B A Tangent any line that contains exactly one point on the circle. BC is tangent to A B A C Point of Tangency B Point of tangency A Common Tangents are lines tangent to more than one coplanar circle. Common External Internal Tangents Tangents X Y B B R A X X B D Tangent Circles are circles that are tangent to each other. Internally Tangent Circles Externally Tangent Circles B B R R A A Sphere is the set of all points in space equidistant from a given point. A B Sphere Radii Diameter Chord Secant Tangent C E B A F D Congruent Circles (or Spheres) WHAT DO THEY HAVE? • have equal radii. B E A D Concentric Circles (or Spheres) share the same center. G Who can think of a real world example? O Q Think , throwing of pointy objects…. Inscribed/Circumscribed A polygon is inscribed in a circle and the circle is circumscribed about the polygon if each vertex of the polygon lies on the circle. L P N M Z Name… 1. 3 radii 2. Diameter 3. Chord R 4. Secant 5. Tangent 6. What is the name for ZX? 7. What should point M be called? O Q X White Boards…. • Draw the following… 1. 2. 3. 4. 5. An inscribed triangle A circle circumscribed about a quadrilateral 2 circles with common external tangents 2 circle that are internally tangent to each other 9.2 Tangents Objectives • Apply the theorems that relate tangents and radii Experiment • Supplies: Pencil, protractor, compass 1. Draw a circle with center A 2. Draw Point B on the bottom of your circle 3. Create line BC tangent to the circle at Point B 4. Draw the radius to the point of tangency and measure the angle formed by the tangent and the radius (L ABC) 5. Compare your measurements with those around you… If the radius and tangent meet at the point of tangency they form a right angle (perpendicular). What can we conclude based on our experiment? A mABC 90 B Point of tangency C X Dunce Cap Rule Tangents to a circle from a common point are congruent. Z Y How do we know the 2 right triangles are congruent? Inscribed/Circumscribed When each side of a polygon is tangent to a circle, the polygon is said to be circumscribed about the circle and the circle is inscribed in the polygon. Each side of the poly, is what to the circle? • GIVEN – Radius = 6 – BC = 8 – Find AC • What allows you to come up with the correct answer? A B • GIVEN – LC = 45 – Radius = 4 – Find AC and BC BC is tangent to C A Whiteboards • Line XY is tangent to D at X. The radius of the circle is 7cm and the length of segment XY is 4cm. FIND THE LENGTH OF SEGMENT DY. Whiteboards • Create a diagram of the following… 1. A triangle circumscribed about a circle 2. A pentagon inscribed inside a circle 9.3 Arcs and Central Angles Objectives • Define and apply the properties of arcs and central angles. Experiment Think – pair - share How is the angle measurement that you just created related to the measurement of a circle? Central Angle Sides – 2 radii Vertex – center of circle L Arc an arc is part of a circle like a segment is part of a line. C A B AC ABC It’s like cutting out a slice of pizza!! This represents the crust of your pizza Central Angle / Arc Measure the measure of an arc is given by the measure of its central angle. (or vise versa) 80 A C 80 m L ABC = 80 mAC 80 B The central angle tells us how much of the 360 ◦ of crust we are using from our pizza. Minor Arc AC < 180 C A B Semicircle • “_____” a circle. • What indicates that we have half a circle? B A ABC C Major Arc ACD > 180 B A C D We only know how to measure angles up to 180, so how do you find the measure of a major arc? Practice Name two minor arcs C R O A S Practice Name two major arcs C R O A S White Board Practice Name two semicircles C R O A S Skip Remember…. • Adjacent angles ? • Angle addition postulate? • Smartboard Skip Adjacent Arcs arcs that have exactly one point in common. D AD A C B DC Arc Addition Postulate D A B mAD mDC mADC C Congruent Central L’s / Arcs Theorem In the same circle or in congruent circles…. • congruent arcs = congruent central angles D E A C B Dissecting a Circle Diagram FREE VISUAL EVIDENCE !!! • Central angles = minor arcs • All the arcs = 360 • Diameters = straight lines = 180 • Vertical angles / adjacent angles H C R O A S White Board Practice H Find the following measurements… C 30 RH AOR HCA 100 50 210 R O 50 A S Pg. 340 • #8 - 13 Section 9-4 Arcs and Chords Objectives • Define the relationships between arcs and chords. REVIEW • WHAT IS A CHORD? • WHAT IS AN ARC? Relationship between a chord and an arc The minor arc between the endpoints of a chord is called the arc of the chord, and the chord between the endpoints of an arc is the chord of the arc. D Chord BD “cuts off” 2 arcs.. BD and BFD B A F Congruent Chords/Arcs Theorem In the same circle or in congruent circles… • congruent arcs have congruent chords • congruent chords have congruent arcs. D If arc BD is congruent to arc FC then… C B A F. Skip - Midpoint/ Bisector of an Arc • Just as we have learned about the bisectors and midpoints of angles and line segments, an arc can be bisected into two congruent arcs D If D is the midpoint of arc BDC, then……. C B A Circle Handout Experiment 1. Label the center A 2. DRAW A CHORD AND LABEL IT DC 3. FIND THE MIDPOINT OF THE CHORD AND LABEL IT X 4. DRAW A RADIUS THAT PASSES THROUGH THE MIDPOINT AND INTERSECTS THE ARC OF THE CIRCLE AT Y 5. USE A PROTRACTOR AND MEASURE LAXC Think – Pair - Share • What facts can you conclude about the arcs, chords, or any other segments? • What is congruent to what? • What about perpendicular? Chord Bisector Theorem A radius or diameter perpendicular to a chord bisects the chord and its arc. D DC BY Y X DX XC C A DY YC B EC: What other 2 segments do I know are congruent that are not explicitly shown? Remember • When you measure distance from a point to a line, you have to make a perpendicular line. A Chord Distance Theorem • In the same circle or in congruent circles: • Chords equally distant from the center are congruent • Congruent chords are equally distant from the center. D E Y B X A C Putting Pythagorean to Work.. Can we use the given information to make a conclusion about the chords shown? (i.e. length of the chords) AY = 3 AX = 3 Radius = 4 D E Y X B A C Hint: Just because something is not shown, doesn’t mean it doesn’t exist! (other radii) THE HEAD BAND PROBLEM: • Find the length of a chord 3cm from the center of a circle with a radius of 7cm. • • • • Workbook page 35 1a 2a Rest on own WHITEBOARDS Solve for x and y D x y C x = 12 y = 12 B 5 A 13 IF arc DB is 55 degrees, then arc CB is? WHITEBOARDS Solve for x and y x=8 y = 16 8 y x WARM - UP 1. What does the term inscribed mean to you in your own words? – Describe the placement of the vertices of an inscribed triangle 2. What do we call the 2 sides and vertex (in circle terms) of a central angle that you learned in 9.3? – What is the measure of the angle equal to? 9.5 Inscribed Angles Objectives • Solve problems and prove statements about inscribed angles. • Solve problems and prove statements about angles formed by chords, secants and tangents. What we’ve learned… • Inscribed means that something is inside of something else – we have looked at inscribed polys and circles • We know that an angle by definition has a vertex and 2 sides that meet at the vertex • In a central angle… – The vertex is the center and the sides are radii Inscribed Angle • Sides – two chords • Vertex – point on the circle A “Intercepted arc” C What do you think the sides are in ‘circle terms’? Where is the vertex? B Experiment 1. Measure the inscribed angle created with a protractor 2. Using the endpoints of the intercepted arc, draw 2 radii to create a central angle and then measure. 3. Compare the measurement of the inscribed angle with that of the central angle measure. 4. Discuss with your partner Theorem inscribed angle = half the measure of the intercepted arc. A mABC mAC ___ 2 C B Corollary If two inscribed angles intercept the same arc, then they are congruent. A ABC ADC D C B Don’t write down, just recgonize WHITEBOARDS • Find the values of r, s, x, y , and z – Take inventory of the diagram before trying to solve! – Concentrate on parts of the whole y◦ r◦ r = 50 s = 50 x = 160 y = 100 z = 100 x◦ O s◦ 80 z◦ Corollary If the intercepted arc is a semicircle, the inscribed angle must = 90. A What is the measure of an inscribed angle whose intercepted arc has the endpoints of the diameter? B O mABC 90 C Corollary Quad inscribed = opposite angles supplementary. mA mC 180 mB mD 180 A B O C D Theorem mABC mADB ___ 2 A B O C D F Treat this angle the same as you would an inscribed angle! Whiteboards • Page 353 – #5, 6, 7 WHITEBOARDS • Find the values of x, y , and z 60◦ X = 30 Y = 60 Z = 150 y◦ O z◦ x◦ 9.6 Other Angles Objectives • Solve problems and prove statements involving angles formed by chords, secants and tangents. WARM UP: •For this diagram…. •Write down the different equations that represent the angle relationships shown. •There’s more than one! 1 4 2 3 Partners: How do you think we can determine the measure of L1? The angle formed by two intersecting chords = half the sum of the intercepted arcs. B m1 (mCB mDE ) _________ C 1 2 m2 (mCE mDB) _________ 2 2 E A D Angles formed from a point outside the circle… 2 secants In each circle, 2 arcs are being intercepted by the angle. 2 tangents 1tangent 1 secant The larger arc is always further away from the vertex. Vertex outside the circle = half the difference of the intercepted arcs. m1 (mBD mEF ) _________ B 2 E C s m a l l 1 F l a r g e A D WHITEBOARDS • ONE PARTNER OPEN BOOK TO PG. 358 • ANSWER #1 – 35 • ANSWER # 6 – 40 • ANSWER # 4 – 80 • ANSWER # 7 – X=50 AB is tangent to circle O. AF is a diameter m AG = 100 B m CE = 30 m EF = 25 8 A C 6 D 30 E 25 3 7 O 2 1 100 G 5 4 F WARM – UP • READ PG. 361 – 363 – Identify what elements are involved in each of the 3 theorems in this section – Example: “Theorem 9-11 refers to the relationship of 2 ___________ intersecting” – What is the idea behind this section…. Angles, segments, circles? ANGLES QUIZ • Identify a numbered angle that represents each of the bullet points. Write an equation representing the measure of the angle •i.e. m L12 = • C D 8 B 4 •X is center of circle E 9 H A 7 6 2 K x 5 G I 3 1 F 1. 2. 3. 4. 5. Central angle Inscribed angle Angle formed inside Angle formed outside Name a 90 angle….