Circles § 11.1 Parts of a Circle § 11.2 Arcs and Central Angles § 11.3 Arcs and Chords § 11.4 Inscribed Polygons § 11.5 Circumference of a Circle § 11.6 Area of a Circle

Vocabulary What You'll Learn Parts of a Circle You will learn to identify and use parts of circles. Vocabulary 1) circle 2) center 3) radius 4) chord 5) diameter 6) concentric

A circle is a special type of geometric figure. Parts of a Circle A circle is a special type of geometric figure. All points on a circle are the same distance from a ___________. center point O B A The measure of OA and OB are the same; that is, OA  OB

Parts of a Circle Definition of a Circle A circle is the set of all points in a plane that are a given distance from a given _____ in the plane, called the ______ of the circle. point center P Note: a circle is named by its center. The circle above is named circle P.

Parts of a Circle There are three kinds of segments related to circles. A ______ is a segment whose endpoints are the center of the circle and a point on the circle. radius A _____ is a segment whose endpoints are on the circle. chord A ________ is a chord that contains the center diameter

that passes through the center. chord Parts of a Circle Segments of Circles radius chord diameter R K J K A K T G From the figures, you can not that the diameter is a special type of _____ that passes through the center. chord

Use Q to determine whether each statement is true or false. Parts of a Circle Use Q to determine whether each statement is true or false. False; C A D B Q Segment AD does not go through the center Q. True; False;

All radii of a circle are _________. congruent Parts of a Circle Theorem 11-1 11-2 All radii of a circle are _________. congruent P R S T G The measure of the diameter d of a circle is twice the measure of the radius r of the circle.

Parts of a Circle Find the value of x in Q. 3x – 5 = 2(17) 3x – 5 = 34 B Find the value of x in Q. 3x – 5 = 2(17) A 3x-5 17 3x – 5 = 34 Q 3x = 39 x = 13 C

P has a radius of 5 units, and T has a radius of 3 units. Parts of a Circle P has a radius of 5 units, and T has a radius of 3 units. R P Q T B A If QR = 1, find RT RT = TQ – QR RT = 3 – 1 RT = 2 If QR = 1, find PQ If QR = 1, find AB PQ = PR – QR AB = 2(AP) + 2(BT) – 1 PQ = 5 – 1 AB = 2(5) + 2(3) – 1 PQ = 4 AB = 10 + 6 – 1 AB = 15

Parts of a Circle P has a radius of 5 units, and T has a radius of 3 units. R P Q T B A If AR = 2x, find AP in terms of x. AR = 2(AP) 2x = 2(AP) x = AP If TB = 2x, find QB in terms of x. QB = 2(TB) QB = 2(2x) QB = 4x

Because all circles have the same shape, any two circles are similar. Parts of a Circle Because all circles have the same shape, any two circles are similar. However, two circles are congruent if and only if (iff) their radii are _________. congruent Two circles are concentric if they meet the following three requirements: They lie in the same plane. S They have the same center. R T They have radii of different lengths. Circle R with radius RT and circle R with radius RS are concentric circles.

Arcs and Central Angles What You'll Learn You will learn to identify major arcs, minor arcs, and semicircles and find the measures of arcs and central angles. Vocabulary 1) central angle 2) arcs 3) minor arc 4) major arc 5) semicircle 6) adjacent arc

Arcs and Central Angles A ____________ is formed when the two sides of an angle meet at the center of a circle. central angle Each side intersects a point on the circle, dividing it into ____ that are curved lines. arcs There are three types of arcs: T S A _________ is part of the circle in the interior of the central angle with measure less than 180°. minor arc central angle R A _________ is part of the circle in the exterior of the central angle. major arc Semicircles __________ are congruent arcs whose endpoints lie on the diameter of the circle.

Arcs and Central Angles Types of Arcs minor arc PG major arc PRG semicircle PRT K T P R G K P G K P G R Note that for circle K, two letters are used to name the minor arc, but three letters are used to name the major arc and semicircle. These letters for naming arcs help us trace the set of points in the arc. In this way, there is no confusion about which arc is being considered.

Arcs and Central Angles Depending on the central angle, each type of arc is measured in the following way. Definition of Arc Measure 1) The degree measure of a minor arc is the degree measure of its central angle. 2) The degree measure of a major arc is 360 minus the degree measure of its central angle. 3) The degree measure of a semicircle is 180.

Arcs and Central Angles In P, find the following measures: = APM P H A M T 46° 80° = 46° APT APT = 80° = 360° – (MPA + APT) = 360° – (46° + 80°) = 360° – (126°) = 234°

Arcs and Central Angles In P, AM and AT are examples of ________ arcs. adjacent P H A M T 46° 80° Adjacent arcs have exactly one point in common. For AM and AT, the common point is __. A Adjacent arcs can also be added. Postulate 11-1 Arc Addition The sum of the measures of two adjacent arcs is the measure of the arc formed by the adjacent arcs. C Q P R If Q is a point of PR, then mPQ + mQR = mPQR

Arcs and Central Angles In P, RT is a diameter. Find mQT. mQT + mQR = mTQR R P S Q T 75° 65° mQT + 75° = 180° mQT = 105° Find mSTQ. mSTQ + mQR + mRS = 360° mSTQ + 75° + 65° = 360° mSTQ + 140° = 360° mSTQ = 220°

Arcs and Central Angles Suppose there are two concentric circles with ASD forming two minor arcs, BC and AD. 60° A D B Are the two arcs congruent? S C Although BC and AD each measure 60°, they are not congruent. The arcs are in circles with different radii, so they have different lengths. However, in a circle, or in congruent circles, two arcs are congruent if they have the same measure.

Arcs and Central Angles Theorem 11-3 In a circle or in congruent circles, two minor arcs are congruent if and only if (iff) their corresponding central angles are congruent. Z Y X W Q WX  YZ iff 60° 60° mWQX = mYQZ

Arcs and Central Angles In M, WS and RT are diameters, mWMT = 125, mRK = 14. Find mRS. WMT  RMS Vertical angles are congruent T W S K M R WMT = RMS Definition of congruent angles mWT = mRS Theorem 11-3 125 = mRS Substitution Find mKS. Find mTS. KS + RK = RS TS + RS = 180 KS + 14 = 125 TS + 125 = 180 KS = 111 TS = 55

Arcs and Central Angles Twenty-two percent of all teens ages 12 through 17 work either full or part-time. Source: ICRs TeenEXCEL survey for Merrill Lynch Teens at Work The circle graph shows the number of hours they work per week. Find the measure of each central angle. 1 – 5: = 72 20% of 360 6 – 10: = 83 23% of 360 11 – 20: = 119 33% of 360 21 – 30: = 50 14% of 360 30 or More: = 36 10% of 360

Vocabulary What You'll Learn Arcs and Chords You will learn to identify and use the relationships among arcs, chords, and diameters. Vocabulary Nothing New!

In circle P, each chord joins two points on a circle. Arcs and Chords In circle P, each chord joins two points on a circle. Between the two points, an arc forms along the circle. By Theorem 11-3, AD and BC are congruent because their corresponding central angles are _____________, and therefore congruent. vertical angles P A C By the SAS Theorem, it could be shown that ΔAPD  ΔCPB. Therefore, AD and BC are _________. congruent B D The following theorem describes the relationship between two congruent minor arcs and their corresponding chords. S

In a circle or in congruent circles, two minor arcs are congruent Arcs and Chords Theorem 11-4 In a circle or in congruent circles, two minor arcs are congruent if and only if (iff) their corresponding ______ are congruent. chords A D AD  BC iff B AD  BC C

The vertices of isosceles triangle ABC are located on R. Arcs and Chords The vertices of isosceles triangle ABC are located on R. If BA  AC, identify all congruent arcs. A BA  AC R C B

Hands-On Arcs and Chords Step 1) Use a compass to draw circle on a G H E F Step 1) Use a compass to draw circle on a piece of patty paper. Label the center P. Draw a chord that is not a diameter. Label it EF. P Step 2) Fold the paper through P so that E and F coincide. Label this fold as diameter GH. Q1: When the paper is folded, how do the lengths of EG and FG compare? EG  FG Q2: When the paper is folded, how do the lengths of EH and FH compare? EG  FG Q3: What is the relationship between diameter GH and chord EF? They appear to be perpendicular.

Arcs and Chords Theorem 11-5 In a circle, a diameter bisects a chord and its arc if and only if (iff) it is perpendicular to the chord. P R D C B A AR  BR and AD  BD iff CD AB Like an angle, an arc can be bisected.

Find the measure of AB in K. Arcs and Chords Find the measure of AB in K. Theorem 11-5 Substitution B C A K D 7

Find the measure of KM in K if ML = 16. Arcs and Chords Find the measure of KM in K if ML = 16. Pythagorean Theorem Given; Theorem 11-5 M K L N 6

Vocabulary What You'll Learn Inscribed Polygons You will learn to inscribe regular polygons in circles and explore the relationship between the length of a chord and its distance from the center of the circle. Vocabulary 1) circumscribed 2) inscribed

Definition of Inscribed Polygon Inscribed Polygons When the table’s top is open, its circular top is said to be ____________ about the square. circumscribed We also say that the square is ________ in the circle. inscribed Definition of Inscribed Polygon A polygon is inscribed in a circle if and only if every vertex of the polygon lies on the circle.

Some regular polygons can be Inscribed Polygons Some regular polygons can be constructed by inscribing them in circles. B A Inscribe a regular hexagon, labeling the vertices, A, B, C, D, E, and F. C P F Construct a perpendicular segment from the center to each chord. From our study of “regular polygons,” we know that the chords AB, BC, CD, DE, and EF are _________ D E congruent From the same study, we also know that all of the perpendicular segments, called ________, are _________. apothems congruent Make a conjecture about the relationship between the measure of the chords and the distance from the chords to the center. The chords are congruent because the distances from the center of the circle are congruent.

In a circle or in congruent circles, two chords are congruent Inscribed Polygons Theorem 11-6 In a circle or in congruent circles, two chords are congruent if and only if they are __________ from the center. equidistant B L M AD  BC iff P A LP  PM C D

In circle O, O is the midpoint of AB. Inscribed Polygons O S T B A C R In circle O, O is the midpoint of AB. If CR = -3x + 56 and ST = 4x, find x definition of midpoint Theorem 11-6 substitution

Circumference of a Circle What You'll Learn You will learn to solve problems involving circumferences of cirlces. Vocabulary 1) circumference 2) pi (π)

Circumference of a Circle An in-line skate advertises “80-mm clear wheels.” The description “80-mm” refers to the diameter of the skates’ wheels. As the wheels of an in-line skate complete one revolution, the distance traveled is the same as the circumference of the wheel. Just as the perimeter of a polygon is the distance around the polygon, the circumference of a circle is the ______________________. distance around the circle

Circumference of a Circle On a sheet of paper, create a table similar to the one below: Circumference Diameter Ratio: C/D Result Example Data 271 86 271 ÷ 86 3.151163 Go to this website to Collect Data from various Circles. Go to this website to Analyze your Data.

Circumference of a Circle In the previous activity, the ratio of the circumference C of a circle to its diameter d appears to be a fixed number slightly greater than 3, regardless of the size of the circle. The ratio of the circumference of a circle to its diameter is always fixed and equals an irrational number called __ or __. pi π Thus, ____ = __, π

Circumference of a Circle Theorem 11-7 Circumference of a Circle If a circle has a circumference of C units and a radius of r units, then C = ____ or C = ___ d r C

Circumference of a Circle

Circumference of a Circle

Vocabulary What You'll Learn Area of a Circle You will learn to solve problems involving areas and sectors of circles. Vocabulary 1) sector

Area of a Circle The space enclosed inside a circle is its area. By slicing a circle into equal pie-shaped pieces as shown below, you can rearrange the pieces into an approximate rectangle. Note that the length along the top and bottom of this rectangle equals the _____________ of the circle, ____. circumference So, each “length” of this approximate rectangle is half the circumference, or __

Area of a Circle The “width” of the approximate rectangle is the radius r of the circle. Recall that the area of a rectangle is the product of its length and width. Therefore, the area of this approximate rectangle is (π r)r or ___.

If a circle has an area of A square units and a radius of Area of a Circle Theorem 11-8 Area of a Circle If a circle has an area of A square units and a radius of r units, then A = ___ r

You could calculate the area if you only knew the radius. Area of a Circle Find the area of the circle whose circumference is 6.28 meters. Round to the nearest hundredth. Use your knowledge of circumference. Solve for radius, r. You could calculate the area if you only knew the radius. Any ideas?

Area of a Sector of a Circle Area of a Circle Theorem 11-9 Area of a Sector of a Circle If a sector of a circle has an area of A square units, a central angle measurement of N degrees, and a radius of r units, then

Area of a Circle