circle: the set of all points in a plane that are equidistant from a given point. center: the point...

Circle: the set of all points in a plane that are equidistant from a given point.

Center: the point from which all points of a circle are set around.

Radius: the distance from the center to a point on the circle.

*two circles are congruent if they have the same radius.

10.1 Tangents to Circles

Diameter: the distance across the circle passing through the center.

*all radii of a circle are congruent. Chord: a segment whose end points are on

the circle. (Diameter is a specific chord) Secant: a line that intersects a circle in

two points. (A chord that keeps going past the sides of the circle)

Tangent: a line in the plane of a circle that intersects the circle in only one point.


Tangent circles: Circles that intersect in one point.

Concentric: circles that share a common center.

Common Tangent: a tangent line that is shared by two coplaner circles.


Interior of a circle: consists of the points that are inside the circle.

Exterior of a circle: consists of the points that are outside the circle.

Point of tangency: the point in which the tangent line intersects with the circle.


Theorem 10.1If a line is tangent to a circle, then it is

perpendicular to the radius drawn to the point of tangency.

If l l is tangent to Q at P, then l QP.

Theorem 10.2In a plane, if a line is perpendicular to a

radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

If l QP, then l is tangent to Q.


Theorem 10.3IF two segments from the same

exterior point are tangent to the circle, then they are congruent.

If SR and ST are tangent to P, then SR ST.


Homework: Page 599 10-16 even, 18-25 all, 26-30 even, 36-38 even, 46-48 even


Central Angle: an angle whose vertex is the center of a circle.

Minor Arc: an arc whose measure is less than 180 .̊

Major Arc: an arc whose measure is greater than 180 ̊.

Measure of an arc: the measure of the central angle of an arc.

Congruent Arcs: two arcs of the same circle or of congruent circles whose measures are congruent.

10.2 Arcs and Chords

Postulate 20 Arc Addition Postulate:The sum of an arc formed by two adjacent

arcs is the sum of the measures of the two arcs.

mABC = mAB + mBC


Theorem 10.4 In the same circle, or in congruent circles, two minor

arcs are congruent if and only if their corresponding chords are congruent.

AB = BC if and only if AB = BC.


Theorem 10.5 If a diameter of a circle is perpendicular to a chord,

then the diameter bisects the chord and its arc.

DE = EF if and only if DG = GF.


Theorem 10.6 If one chord is a perpendicular bisector of another

chord, then the first chord is a diameter.

JK is a diameter of the circle.


Theorem 10.7 In the same circle, or in congruent circles, two

chords are congruent if and only if they are equidistant from the center.

AB = CD if and only if EF = EG.


Homework: Page 607 12-46 even


Inscribed angles: an angle whose vertex is on a circle and whose sides contain chords of the circle.

Intercepted arc: the arc that lies in the interior of an inscribed angle and has endpoints on the angle.

Inscribed: all of the vertices of a polygon lie on the circle

Circumscribed: the circle is around the polygon.

10.3 Inscribed Angles

Theorem 10.8 Measure of an inscribed angle

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

M ADB = ½ mAB


Theorem 10.9 If two inscribed angles of a circle

intercept the same arc, then the angles are congruent.


Theorem 10.10If a right triangle is inscribed in a circle,

then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.

B is a right angle if and only if AC is a diameter of the circle.


Theorem 10.11A quadrilateral can be inscribed in a

circle if and only if its opposite angles are supplementary.

D, E, F, and G lie on some circle, C, if and only if m D + m F = 180 .̊


Theorem 10.12If 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 intersected arc.

M 1 = ½ mABM 2 = ½ mBCA

10.4 Other Angle Relationships in Circles

Theorem 10.13If 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 verticle angle

M 1 = ½ (mCD + mAB), M 2 = ½ (mBC + mAD)


Theorem 10.14IF 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.

m 1 = ½ (mBC - mAC) m 2 = ½ (mPQR - mPR) m 3 = ½(mXY – mWZ)


Theorem 10.15If 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 product of the lengths of the segments of the other chord.

10.5 Segment Lengths in Circles

Tangent segment: a segment that is tangent to the circle at the segments endpoint.

Secant Segment: a segment that passes through the circle, but still ends on the side of the circle.

External segment: the portion of the secant segment that falls outside the circle.


Theorem 10.16If two secant segments share the same

endpoint outside the 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 the external segment.


Theorem 10.17If 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.


circle: the set of all points in a plane that are equidistant from a given point. center: the point...

Documents

circle tangent

tangent circles

congruent circles

tangent line

congruent arcs

common tangent

major arc

intercepted arc