ssrs barcode font Concentric circles are circles that have the same center. in Objective-C

Drawer Quick Response Code in Objective-C Concentric circles are circles that have the same center.

Concentric circles are circles that have the same center.
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Fig. 6-4
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Fig. 6-5
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Thus, the two circles shown in Fig. 6-5 are concentric circles. AB is a tangent of the inner circle and a chord of the outer one. CD is a secant of the inner circle and a chord of the outer one.
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Two circles are equal if their radii are equal in length; two circles are congruent if their radii are congruent. Two arcs are congruent if they have equal degree measure and length. We use the notation mAC to denote measure of arc AC.
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6.1A Circle Principles
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PRINCIPLE
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A diameter divides a circle into two equal parts.
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Thus, diameter AB divides circle O of Fig. 6-6 into two congruent semicircles, ACB and ADB.
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CHAPTER 6 Circles
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PRINCIPLE
If a chord divides a circle into two equal parts, then it is a diameter. (This is the converse of Principle 1.)
Thus if ACB > ADB in Fig. 6-6, then AB is a diameter.
Fig. 6-6
PRINCIPLE 3:
A point is outside, on, or inside a circle according to whether its distance from the center is greater than, equal to, or smaller than the radius.
F is outside circle O in Fig. 6-6, since FO is greater in length than a radius. E is inside circle O since EO is smaller in length than a radius. A is on circle O since AO is a radius.
Fig. 6-7
Fig. 6-8
PRINCIPLE
Radii of the same or congruent circles are congruent.
Thus in circle O of Fig. 6-7, OA > OC.
PRINCIPLE
Diameters of the same or congruent circles are congruent.
Thus in circle O of Fig. 6-7, AB > CD.
PRINCIPLE
In the same or congruent circles, congruent central angles have congruent arcs.
Thus in circle O of Fig. 6-8, if j 1 > j 2, then AC > CB.
PRINCIPLE
In the same or congruent circles, congruent arcs have congruent central angles.
Thus in circle O of Fig. 6-8, if AC > CB, then j 1 > j 2. (Principles 6 and 7 are converses of each other.)
PRINCIPLE
In the same or congruent circles, congruent chords have congruent arcs.
Thus in circle O of Fig. 6-9, if AB > AC, then AB > AC.
CHAPTER 6 Circles
9: In the same or congruent circles, congruent arcs have congruent chords.
PRINCIPLE
Thus in circle O of Fig. 6-9, if AB > AC, then AB > AC. (Principles 8 and 9 are converses of each other.)
Fig. 6-9
Fig. 6-10
PRINCIPLE 10:
A diameter perpendicular to a chord bisects the chord and its arcs.
Thus in circle O of Fig. 6-10, if CD ' AB, then CD bisects AB, AB, and ACB. A proof of this principle is given in 16.
PRINCIPLE 11:
A perpendicular bisector of a chord passes through the center of the circle.
Thus in circle O of Fig. 6-11, if PD is the perpendicular bisector of AB, then PD passes through center O.
Fig. 6-11
PRINCIPLE
Fig. 6-12
12: In the same or congruent circles, congruent chords are equally distant from the center.
Thus in circle O of Fig. 6-12, if AB > CD, if OE ' AB, and if OF ' CD, then OE > OF.
PRINCIPLE
13: In the same or congruent circles, chords that are equally distant from the center are congruent.
Thus in circle O of Fig. 6-12, if OE > OF, OE ' AB, and OF ' CD, then AB > CD. (Principles 12 and 13 are converses of each other.)
SOLVED PROBLEMS
6.1 Matching test of circle vocabulary Match each part of Fig. 6-13 on the left with one of the names on the right: (a) OE (b) FG 1. Radius 2. Central angle
CHAPTER 6 Circles
Fig. 6-13
(c) FH (d) CD (e) IJ (f) EF (g) FGH (h) FEG (i) j EOF (j) Circle O about EFGH (k) Circle O in ABCD (l) Quadrilateral EFGH (m) Quadrilateral ABCD
Solutions
(a) 1 (b) 6 (c) 7 (d) 9 (e) 8 (f) 4 (g) 3 (h) 5
3. Semicircle 4. Minor arc 5. Major arc 6. Chord 7. Diameter 8. Secant 9. Tangent 10. Inscribed polygon 11. Circumscribed polygon 12. Inscribed circle 13. Circumscribed circle
(i) 2 (j) 13
(k) 12 (l) 10
(m) 11
6.2 Applying principles 4 and 5 In Fig. 6-14, (a) what kind of triangle is OCD; (b) what kind of quadrilateral is ABCD (c) In Fig. 6-15 if circle O circle Q, what kind of quadrilateral is OAQB
Solutions
Radii or diameters of the same or equal circles have equal lengths. (a) Since OC > OD, ^ OCD is isosceles.
Fig. 6-14
Fig. 6-15
CHAPTER 6 Circles
(b) Since diagonals AC and BD are equal in length and bisect each other, ABCD is a rectangle. (c) Since the circles are equal, OA > AQ > QB > BO and OAQB is a rhombus.
6.3 Proving a circle problem Given: AB > DE BC > EF
To Prove: j B > j E Plan: Prove ^ I > ^ II.
PROOF:
Statements 1. AB > DE, BC > EF 2. AB > DE, BC > EF 3. ABC > DEF 4. AC > DF 5. ^ I > ^ II 6. j B > j E 1. Given
Reasons
2. In a circle, > chords have > arcs. 3. If equals are added to equals, the sums are equal. Definition of > arcs. 4. In a circle, > arcs have > chords. 5. SSS
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