Circles Vocabulary Unit 7

OBJECTIVES: • Degree & linear measure of arcs• Measures of angles in circles• Properties of chords, tangents, & secants

About Circles• Definition: set of coplanar points equidistant from a

given point P(center) written P • Chord: any segment having endpoints on the circle• Radius (r): a segment from a point on the circle to the

center• Diameter (d): chord containing the center of the circle• Circumference: the distance around the circle

Circumference: C = πd = 2πr• Concentric circles share the same center & have

different radius lengths

Angles and Arcs Measure• Central angles have the vertex at the center of the circle• The sum of non-overlapping central angles = 360°• A central angle splits the circle into 2 arcs:

minor arc: m major arc: m

• Adjacent arcs share only the same radius The measure of 2 adjacent arcs can be added to form one bigger arc.

Arc Length is the proportion of the circumference formed by the central angle :

L

T

V . P

TL --has same degree as central LVT -- degree = 360 central

central C360

Arcs and Chords

-Two minor arcs are iff their corr chords are- Inscribed polygons has each vertex on the circle- If the diameter of a circle is perpendicular to a

chord, it bisects the cord & the arc

-Two chords are iff they are equidistant from the center.

arc of the chord chord

11

11

.

Inscribed Angles

• An inscribed has its vertex on the circle• Inscribed polygons have all vertices on the circle• Opposite ‘s of inscribed quadrilaterals are

supplementary

• The measure of inscribed ’s = ½ intercepted arc• If an inscribed intercepts a semicircle, the = 90°• If 2 inscribed ‘s intercept the same arc, the ‘s are

red & blue ‘s are

Inscribed Intercepted arc

TangentsTangent lines intersect the circle at 1

point—the ‘point of tangency’

• A line is tangent to the circle iff it is perpendicular the the radius drawn at that particular point

•

• if a point is outside the circle & 2 tangent segments are drawn from it, the 2 segments are congruent.

• Tangents can be internal or external

.

Secants, Tangents & Angle Measures

A secant line intersects the circle in 2 points

A B

C 1ABC = 2BCD

1BCD = 2BD

Central angles1 secant & 1 tangent

I

intersecting at point of tangency

Secants, Tangents & Angle Measures

2 secants: forms 2 pair of vertical angles –

vertical

II

A

B C

D

12

1 1 = ( AB + AD)21 2 = ( AB + CD)2

m m m

m m m

intersection in interior of circle

's

Secants, Tangents & Angle Measures

Case 1 2 secants

III

Intersection at exterior point

P

Case 2 1 secant & 1 tangent

Case 3 2 tangents

P

P

A

B C

D

AB

C D

A

B

1CPD ( CD AB)2

m m m

1CPB ( CB AD)2

m m m

Q 1APB ( AQB AB)2

m m m

Special Segments in a CircleIf two chords intersect inside (or outside)

of a circle, the products of their segments are equal ab = cd

2 secants & exterior point::

a(a + x) = b(b + c)

abc

d

xa

b c

1 tan and 1 sec & exterior pointa

x ba2 = x(x + b)

= x2 + bx