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# Circle geometry

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### Circle geometry

1. 1. CIRCLE GEOMETRY Jyoti Vaid
2. 2. Circles  11.1 Parts of a Circle  11.4 Inscribed Polygons  11.3 Arcs and Chords  11.2 Arcs and Central Angles  11.6 Area of a Circle  11.5 Circumference of a Circle
3. 3. 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
4. 4. 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 centerDiameter
5. 5. Parts of a Circle Segments of Circles K A diameterchordradius R K J K T G Kof radiusaisKA Kof chordaisJR Kof diameteraisGT From the figures, you can not that the diameter is a special type of CHORD that passes through the center.
6. 6. Theorem 11-1 Theorem 11-2 Parts of a Circle All radii of a circle are Congruent . P R S T G PTPSPGPR The measure of the diameter d of a circle is twice the measure of the radius r of the circle. r or d rd 2 1 2
7. 7. S 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: Circle R with radius RT and circle R with radius RS are concentric circles.  They lie in the same plane.  They have the same center.  They have radii of different lengths. R T
8. 8. T S Arcs and Central Angles A ____________ is formed when the two sides of an angle meet at the center of a circle. central angle R central angle Each side intersects a point on the circle, dividing it into arcs that are curved lines. There are three types of arcs: A _________ is part of the circle in the interior of the central angle with measure less than 180 . minor arc A _________ is part of the circle in the exterior of the central angle. major arc __________ are congruent arcs whose endpoints lie on the diameter of the circle. Semicircles
9. 9. K P G R Arcs and Central Angles Types of Arcs semicircle PRTmajor arc PRGminor arc PG K P G 180PGm 180PRGm 180PRTm 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. K T P R G
10. 10. 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.
11. 11. Arcs and Central Angles In P, find the following measures: P H AM T 46° 80° MAm = APM MAm = 46 ATmAPT THMm = 360 – ( MPA + APT) APT = 80 THMm = 360 – (46 + 80 ) THMm = 360 – (126 ) THMm = 234
12. 12. Arcs and Central Angles P H AM T 46° 80° In P, AM and AT are examples of Adjacent Arcs . 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 Postulate The sum of the measures of two adjacent arcs is the measure of the arc formed by the adjacent arcs. C QP R If Q is a point of PR, then mPQ + mQR mPQR
13. 13. Arcs and Central Angles B Suppose there are two concentric circles S60° C A D with ASD forming two minor arcs, BC and AD. Are the two arcs 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. Although BC and AD each measure 60 , they are not congruent.
14. 14. 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. 60° 60° Z Y X W Q WX YZ iff m WQX = m YQZ
15. 15. Arcs and Central Angles T W S K M R In M, WS and RT are diameters, m WMT = 125, mRK = 14. Find mRS. WMT RMS Vertical angles are congruent WMT = RMS Definition of congruent angles mWT = mRS Theorem 11-3 125 = mRS Substitution Find mKS. KS + RK = RS KS + 14 = 125 KS = 111 Find mTS. TS + RS = 180 TS + 125 = 180 TS = 55
16. 16. Arcs and Chords BD S C P A In circle P, each chord joins two points on a circle. Between the two points, an arc forms along the circle. vertical angles By Theorem 11-3, AD and BC are congruent because their corresponding central angles are _____________, and therefore congruent. By the SAS Theorem, it could be shown that ΔAPD ΔCPB. Therefore, AD and BC are _________.congruent The following theorem describes the relationship between two congruent minor arcs and their corresponding chords.
17. 17. 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. B D C A chords AD BC iff AD BC
18. 18. Arcs and Chords A BC The vertices of isosceles triangle ABC are located on R. R If BA AC, identify all congruent arcs. BA AC
19. 19. Arcs and Chords 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. Step 2) Fold the paper through P so that E and F coincide. Label this fold as diameter GH. E F P G H Q1: When the paper is folded, how do the lengths of EG and FG compare? Q2: When the paper is folded, how do the lengths of EH and FH compare? Q3: What is the relationship between diameter GH and chord EF? EG FG EG FG They appear to be perpendicular.
20. 20. 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.
21. 21. Arcs and Chords B C A K D 7 Find the measure of AB in K. DBAB 2 Theorem 11-5 72AB 14AB Substitution
22. 22. Arcs and Chords M K L K N6 Find the measure of KM in K if ML = 16. 222 MNKNKM Pythagorean Theorem 222 86KM Given; Theorem 11-5 6436 2 KM 100 2 KM 100 2 KM 10KM
23. 23. Inscribed Polygons A F B C D E Some regular polygons can be constructed by inscribing them in circles. Inscribe a regular hexagon, labeling the vertices, A, B, C, D, E, and 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 _________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. P
24. 24. 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 L M P C D B A AD BC iff LP PM
25. 25. 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, ____ = __, d C π .or dC .aswrittenbealsocaniprelationshthe2r,dSince rC 2
26. 26. 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 = ____r2 or C = ___d C d r
27. 27. 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 a approximate rectangle. Note that the length along the top and bottom of this rectangle equals the _____________ of the circle, ____.circumference r2 So, each “length” of this approximate rectangle is half the circumference, or __r
28. 28. 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 ___. 2 r
29. 29. 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 = ___ 2 r 2 rA r
30. 30. Sectors are a fractional part of a circle’s area
31. 31. Find the shaded area 8 A = πr² A = 64π Sector area = ¼ of 64π 64π = 16π 4 90 of circle’s area 360
32. 32. 60° 60 of circle’s area 360 12 Area = πr² A = 144π Sector area = A = 1 of 144π 6 144π = 24π 6
33. 33. AREA OF SECTION = AREA OF SECTOR – AREA OF TRIANGLE ¼ π r² - ½ bh
34. 34. Area of section = area of sector – area of triangle ¼ π r² - ½ bh 10 A OF = ½∙10∙10= 50 A OF SECTION = 25π - 50 A of circle = 100π A OF = ¼ 100π = 25π
35. 35. 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 2 360 r N A
36. 36. Inscribed Angle: An angle whose vertex is on the circle and whose sides are chords of the circle
37. 37. Determine whether each angle is an inscribed angle. Name the intercepted arc for the angle. C L O T 1. YES; CL
38. 38. Determine whether each angle is an inscribed angle. Name the intercepted arc for the angle. Q R K V 2. NO; QVR S
39. 39. 2 ArcdIntercepte AngleInscribed 1600 800 To find the measure of an inscribed angle…
40. 40. 120 x What is the value of x? y What do we call this type of angle? How do we solve for y? The measure of the inscribed angle is HALF the measure of the inscribed arc!!
41. 41. 72 If two inscribed angles intercept the same arc, then they are congruent.
42. 42. If all the vertices of a polygon touch the edge of the circle, the polygon is INSCRIBED and the circle is CIRCUMSCRIBED.
43. 43. A circle can be circumscribed around a quadrilateral if and only if its opposite angles are supplementary. A B CD 180CmAm 180DmBm
44. 44. z y 110 85 110 + y =180 y = 70 z + 85 = 180 z = 95 Example 8 Find y and z.
45. 45. 180 If a right triangle is inscribed in a circle then the hypotenuse is the diameter of the circle.
46. 46. H K G N 4x – 14 = 90 Example 6 In K, m<GNH = 4x – 14. Find the value of x. x = 26