Circle GEOMETRY

Radius (or Radii for plural)

O

A The segment joining the center of a circle to a point on the circle.

Example: OA

Diameter

A chord that passes through the center of a circle.

Example: AB

O

A

B

Chord

B

CA

A segment joining two points on a circle

Example: AB

Chord

B

C

A

A segment joining two points on a circle

Example: AB

Secant

A line that intersects the circle at exactly two points.

Example: ABD

C

B

A

O

Tangent

C

B

A

A line that intersects a circle at exactly one point.

Example: AB

Arc

A figure consisting of two points on a circle and all the points on the circle needed to connect them by a single path.

Example: arc AB

A

B

Diagram of Arcs

CD B

Aminor arc: AB

major arc: ABD

Measure of a minor arc – the measure of its central angleMeasure of a major arc – the difference between 360° and the measure of its associated minor arc.

Central Angle

An angle whose vertex is at the center of a circle.

Example: Angle ABC

A

B

C

Central angles will always equal the inscribed arc.– Example: angle ABC

= arc AC

A

B

C

Example 2 Find the measures of the red arcs. Are the arcs congruent?

41

41

AC

D

EmAC = mDE = 41Since the arcs are in the same circle, they are congruent!

Example 1

Find the measure of each arc.

70PN L

M

a. LM

c. LMN

b. MNL

70°

360° - 70° = 290°

180°

Inscribed Angle

AB

C

An angle whose vertex is on a circle and whose sides are determined by two chords.

Example: Angle ABC

Intercepted Arc

AB

C

An arc that lies in the interior of an inscribed angle.

Example: arc AC

An inscribed arc will always equal twice the inscribed angle.– Ex. Arc AC= 2 times

Angle ABC

AB

C

Example 1 Find the measure of the blue arc or angle.

RS

QT

a.

mQTS = 2(90 ) = 180

b.80

E

FG

mEFG = 1

2(80 ) = 40

Example 1

Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius.

FC

B

G

A

H

D

E

Id. CE

c. DF

b. EI

a. AH tangent

diameter

chord

radius

Tangent Theorem

The tangent is a line or line segment that touches the perimeter of a circle at one point only and is perpendicular to the radius that contains the point. 

Example 3

Tell whether CE is tangent to D.

45

43

11

D

E

C

Use the converse of the Pythagorean Theorem to see if the triangle is right.

112 + 432 ? 452

121 + 1849 ? 2025

1970 2025

CED is not right, so CE is not tangent to D.

Definitions

Inscribed polygon – a polygon whose vertices all lie on a circle.

Circumscribed circle – A circle with an inscribed polygon.

The polygon is an inscribed polygon and the circle is a circumscribed circle.

Inscribed Quadrilateral

If a quadrilateral is inscribed in a circle, then both pairs of opposite angles are supplementary.

1. Problem: Find the measure of arc GDE.

Solution: By the theorem stated above, angle D and angle F are supplementary. Therefore, angle F equals 95o. The first theorem discussed in this section tells us the measure of an arc is twice that of its inscribed angle. With that theorem, arc GDE is 190o.

Inscribed Right Triangle Theorem

If 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. A

C

BB is a right angle if and only if ACis a diameter of the circle.

Example 3 Find the value of each variable.

2x

Q

A

B

C

a.

2x = 90

x = 45

b. z

y

80

120

D

E

F

G

mD + mF = 180

z + 80 = 180

z = 100

mG + mE = 180

y + 120 = 180

y = 60

Chord Product Theorem

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 product of the lengths of the segments of the other chord.

E

C

D

A

B

EA EB = EC ED

Example 1

Find the value of x.

x

96

3

E

B

D

A

C3(6) = 9x

18 = 9x

x = 2

Try This!

Find the value of x.x 9

18

12

E

B

D

A

C

9(12) = 18x

108 = 18x

x = 6