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Circle. Is the set of all points equidistant from a given point called the center. The man is the center of the circle created by the shark. Parts of a circle. We name a circle using its center point. ⊙ W is shown W •

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Circle

Circle

Is the set of all points equidistant from a given point called the center.

The man is the center of the circle created by the shark.


Parts of a circle
Parts of a circle

We name a circle using its center point.

⊙ W is shown

W•

So, the Center is defined as the point inside the circle equidistant from all points on the circle.


Diameter: Any line segment (chord) that contains the circle’s center. BE (ALL diameters of a given circle are  )Radius: Any line segment with 1 endpoint at the center of the circle, and the other endpoint ON the circle. XF (The Radius is = ½ diameter. All radii of a given circle are ) XF  XE  XB

B .

. X

.E

F


Chord: Any line circle’s center. BE segment with endpoints that lie on the circle.CD Secant: Any line that intersects the circle in exactly 2 points (cuts through).CDTangent: Line, segment or ray that intersects the circle in exactly 1 point (touches)YA or YAAlways perpendicular to radius with endpoint at point of tangency. XA  YA

C.

Y .

.D

. X

A .


Arcs – minor arc circle’s center. BE (less than ½ the circle) AB MB YXA major arc (at least half the circle) BMY BAM XYBSector - area formed by two radii and the arc formed by them (green area) – like a slice of pizza.

M

C

E Y

B

X

A


ANGLES in a circle circle’s center. BE Central angle: center of circle as vertex and radii as sides.  ACBInscribed angle: vertex point on the circle and chords as sides.  AMBExterior angle: vertex outside the circle, either secants or tangents as sides.  E or  AEBInterior angle: vertex is intersection of two chords inside the circle (not the center).  AWB

m  ACB = m AB m  AEB = ½(m AB – m XY)

m  AMB = ½ m AB m  AWB = ½ (m AB + m MY)

M

C

E Y W

B

X

A


Segment lengths in a circle exterior angle forms inverse proportions note order
SEGMENT Lengths in a circle circle’s center. BE Exterior angle forms inverse proportions (* NOTE ORDER)

PR = PS PR • PQ = PT • PS

PT PQ

P

S

T

Q

R


SEGMENT Lengths in a circle circle’s center. BE related to chords, secants and tangentsExterior angles form proportions (* NOTE ORDER)

FG = FJ FG • FH = FJ • FJ

FJ FH

F G H

J


SEGMENT Lengths in a circle circle’s center. BE related to chords, secants and tangentsInterior angles form proportions (* NOTE ORDER)

BE = AE BE • ED = AE • BC

EC ED

A

D

E

B

C


Postulates and theorems about circles
Postulates and theorems about circles circle’s center. BE

  • The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

  • In the same or congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.


Postulates and theorems about circles1
Postulates and theorems about circles circle’s center. BE

  • If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

  • If one chord is perpendicular bisector of another chord then the first chord is a diameter.

  • In the same or congruent circles, two chords are congruent if and only if they are equidistant from the center.


Postulates and theorems about circles2
Postulates and theorems about circles circle’s center. BE

  • If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.

  • If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

  • A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.


Postulates and theorems about circles3
Postulates and theorems about circles circle’s center. BE

  • If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

  • If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

  • If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.


Postulates and theorems about circles4
Postulates and theorems about circles circle’s center. BE

  • If two chords intersect in the interior of a circle then the product of the lengths of the segments of one chord is equal to the p;product of the lengths of the segments of the other chord.

  • If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment.


Postulates and theorems about circles5
Postulates and theorems about circles circle’s center. BE

  • If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.


Postulates and theorems about circles6
Postulates and theorems about circles circle’s center. BE

  • In a circle of radius r, an arc of degree measure m has arc length equal to (m/360 • 2πr).

  • In a circle of radius r, where a sector has an arc degree measure of m, the area of the sector is (m/360 •πr2)

  • The area of a circle is πr2

  • The circumference of a circle is 2πr


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