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.

With your compass… Create a circle on your notes that is about half the page As we go through the basic terms for a circle label each one on your notes

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

The Circle The given distance is a radius (plural radii) How many radii does a circle have? Are they all the same length? B radius A

The Circle A The given point is the center B radius A center “Circle with center A”

The Circle What are some real world examples of circles? B Point on circle A

What is the difference between a line and a line segment? Think – Pair - Share What is the difference between a line and a line segment?

Chord any segment whose endpoints are on the circle. C chord B A

Diameter A chord that contains the center of the circle B A What is another name for half of the diameter? diameter C

Secant any line that contains a chord of a circle. C secant B A

Tangent any line that contains exactly one point on the circle. BC is tangent to A B A tangent C

Point of Tangency B Point of tangency A

Common Tangents are lines tangent to more than one coplanar circle. Common External Tangents Common Internal Tangents Y B X B R R A 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. B A

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 O Who can think of a real world example? Think , throwing of pointy objects…. Q

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.

M Q O N R L P Z Name… 3 radii Diameter Chord Secant Tangent What is the name for ZX? What should point M be called? X

White Boards…. Draw the following… 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 Draw a circle with center X Draw Point Y on the bottom of your circle Create line ZY tangent to the circle at Point Y Draw the radius to the point of tangency and measure the angle formed by the tangent and the radius (L XYZ) Compare your measurements with those around you…

Theorem If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. B What can we conclude based on our experiment? A tangent C

Theorem (fill in the blank) If a line in the plane of a circle is perpendicular to a radius at its endpoint, then the line is a tangent to the circle. B tangent A X

How do we know the 2 right triangles are congruent? X Dunce Cap Rule Tangents to a circle from a common point are congruent. tangent tangent Z Y A 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 BC is tangent to A Radius = 6 BC = 8 Find AC What allows you to come up with the correct answer? LC = 45 BC = 4 A B C BC is tangent to A

Whiteboards Create a diagram of the following… A triangle circumscribed about a circle A pentagon inscribed inside a circle

9.3 Arcs and Central Angles Objectives Define and apply the properties of arcs and central angles.

Experiment Draw a straight line using a straightedge Extend the line so that its sides go past the outside of your protractor Connect the two sides of your angle using the outer curved side of your protractor What do you end up with? What do I get with a 70 degree angle?

Think – pair - share How is the angle measurement that you just created related to the measurement of a circle?

Central Angle is formed by two radii, with the center of the circle as the vertex. It’s like cutting out a slice of pizza!! Arc C an arc is part of a circle like a segment is part of a line. A L This represents the crust of your pizza B

Central Angle / Arc Measure the measure of an arc is given by the measure of its central angle. (or vise versa) 80 A C The central angle tells us how much of the 360 ◦ of crust we are using from our pizza. 80 m L = 80 B

Minor Arc an unbroken part of a circle with a measure less than 180°.

Semicircle an unbroken part of a circle that shares endpoints with a diameter. How do I know that AC is a diameter? B A C

Major Arc an unbroken part of a circle with a measure greater than 180°. We only know how to measure angles up to 180, so how do you find the measure of a major arc? B A C D

Practice Name two minor arcs C R O A S AR, RC, RS, AS, SC

Practice Name two major arcs C R O A S

White Board Practice Name two semicircles C R O A S

Angle addition postulate? Skip Remember…. Adjacent angles ? Angle addition postulate? Smartboard

Skip Adjacent Arcs arcs that have exactly one point in common. D A C B

Arc Addition Postulate B C

Theorem In the same circle or in congruent circles…. congruent arcs = congruent central angles 90 D 90 90 90 A C B smartboard

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 Find the following measurements… H C 30 100 50 210 R RH AOR HCA O 50 A S

Group Practice Give the measure of each arc. D C 2x-14 4x 2x B E 3x

Group Practice m AB = 88 m BC = 52 m CD = 38 m DE = 104 m EA = 78 D C 2x-14 4x 2x B E 3x 3x + 10 A

The radius of the Earth is about 6400 km.

The latitude of the Arctic Circle is 66. 6º North The latitude of the Arctic Circle is 66.6º North. That means the m BE 66.6º. B A 66.6º 6400 6400 W E O

Find the radius of the Arctic Circle xº B A 66.6º 6400 W E O

Find the radius of the Arctic Circle 23.4º B A 66.6º 6400 W E O

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 A F B

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 A B

Circle Handout Experiment Label the center A DRAW A CHORD AND LABEL IT DC FIND THE MIDPOINT OF THE CHORD AND LABEL IT X DRAW A RADIUS THAT PASSES THROUGH THE MIDPOINT AND INTERSECTS THE ARC OF THE CIRCLE AT Y 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?

Theorem A diameter (or radius) that is perpendicular to a chord bisects the chord and its arc. D Y X C A 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

Putting Pythagorean to Work.. Partners: Use the given information do make a conclusion about the chords shown. Hint: Just because something is not shown, doesn’t mean it doesn’t exist! (other radii) AY = 3 AX = 3 E D Y X B C A 4

Theorem In the same circle or in congruent circles: . Chords equally distant from the center (or centers) are congruent Congruent chords are equally distant from the center (or centers) . E D Y B X C A

WHITEBOARDS Solve for x and y x = 12 y = 12 C x = 12 y = 12 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

Whiteboards Find the length of a chord 3cm from the center of a circle with a radius of 7cm.

WARM - UP What does the term inscribed mean to you in your own words? Describe the placement of the vertices of an inscribed triangle 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

Create a circle using a compass & an inscribed angle in your notes An angle formed by two chords or secant lines whose vertex lies on the circle. A “Intercepted arc” C What do you think the sides are in ‘circle terms’? Where is the vertex? Create a circle using a compass & an inscribed angle in your notes B

Experiment Measure the inscribed angle created with a protractor Using the endpoints of the intercepted arc, draw 2 radii to create a central angle and then measure. Compare the measurement of the inscribed angle with that of the central angle measure. Discuss with your partner

Theorem The measure of an inscribed angle is half the measure of the intercepted arc. A ___ 2 C B

Corollary If two inscribed angles intercept the same arc, then they are congruent. A D C B Don’t write down, just recgonize

WHITEBOARDS r = 50 s = 50 x = 160 y = 100 z = 100 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 80 s◦ 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 C

Corollary If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. A B O D C

Treat this angle the same as you would and inscribed angle! Theorem ___ 2 A B O C D F Treat this angle the same as you would and inscribed angle!

Whiteboards Page 353 #7, 6

X = 30 Y = 60 Z = 150 WHITEBOARDS Find the values of x, y , and z 60◦

9.6 Other Angles Objectives Solve problems and prove statements involving angles formed by chords, secants and tangents.

1 2 4 3 WARM UP: Draw a central angle and label it Draw an inscribed angle and label it FOR BOTH CREATE AN EQUATION FOR THEIR MEASURES IN COMPARISON TO THE INTERCEPTED ARC 1 2 4 WARM UP: For this diagram…. Write down the different equations that represent the angle relationships shown. There’s more than one! 3

Theorem _________ 2 _________ 2 Partners: How do you think we can determine the measure of L1? Theorem The angle formed by two intersecting chords is equal to half the sum of the intercepted arcs. B _________ 2 C 1 A 2 _________ 2 E D

Angles formed from a point outside the circle… 2 secants 2 tangents In each circle, 2 arcs are being intercepted by the angle. 1tangent 1 secant The larger arc is always further away from the vertex.

Theorem The angle formed by secants or tangents with the vertex outside the circle has a measure equal to half the difference of the intercepted arcs. B E C large small 1 A F _________ 2 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 AB is tangent to circle O. AF is a diameter m AG = 100 m CE = 30 m EF = 25 B D C 6 30 E 8 25 O 3 5 A F 7 2 1 4 100 G

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?

9.7 Circles and Lengths of Segments Objectives Solve problems about the lengths of chords, secants and tangents.

Skip - Open Books to Pg. 361 Read the paragraphs for section 9-7 “segments of a chord” AP and PB D A P O C B

Theorem If a chord intersects another chord each chord now has how many parts? If you multiply those 2 parts together they will equal the product of the other chords 2 parts. B E X F D

Example Solve for x 2 x • 12 = 3 • x 24 = 3x 8 = x 3 12

Skip - Pg. 362 Read the middle paragraph “external segments” aka “outside piece” BP and DP “Secant segments” Aka “whole piece” AP and CP D B O C A

Theorem 2 Secants Whole  Outside Part = Whole  Outside Part Each one is made up of an outside part and an inside part that make the whole. B E C A F D Whole  Outside Part = Whole  Outside Part

Example Whole  Outside Part = Whole  Outside Part 8 8 •x = 12 •4 8x = 48 x = 6 x 4 8

Skip Pg. 362 and 363 Read the bottom of 362 and top of 363 “external segments” BP and PD “Secant segments” AP and PD P B A D Whole  Outside Part = Whole  Outside Part

Theorem 1 Secant 1 Tangent Whole  Outside Part = Whole  Outside Part Still the same parts… outside + inside = whole 1 Tangent Only made up of an outside part…which is also the whole!! E C A F D Whole  Outside Part = Whole  Outside Part

Example Whole Piece Outside Piece = Whole Piece Outside Piece x •x = 24 •6 x2 = 144 x = 12 6 18 x

WHITEBOARDS ONE PARTNER OPEN BOOK TO PG. 363/364 ANSWER #1 ANSWER # 4 x = 8 ANSWER # 4 x = 4 ANSWER # 5

WHITEBOARDS 5 4 x 3 y X = 9 Y = 6

ANGLES QUIZ C D 8 B 4 E H 2 7 K 3 A x 5 6 1 F G I 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 E H 2 7 K 3 A x 5 6 1 Central angle Inscribed angle Angle formed inside Angle formed outside 90 angle F G I