Geometry Notes: Circles and Arcs



Geometry Notes C – 1: Circles and Arcs

Circles

A circle is the set of points in a plane that are

A circle is usually named by its center: Circle C

Facts: All radii of a given circle are

Two circles are congruent if

Central Angles

A central angle of a circle is any angle

Arcs

An arc of a circle is

A minor arc is any arc that is

A major arc is any arc that is

A semicircle is an arc that is

The degree measure of an arc is

Note: degree measure is NOT the same as arc length.

Facts: A full circle is

A semicircle is

Congruent arcs are two arcs of the same circle (or congruent circles) that have

Congruent central angles intercept congruent angles and vice versa.

Arcs that share an endpoint but no other points may be added.

Ex: In circle O, [pic] and [pic]. Find

a. [pic]

b. [pic]

c. [pic]

Ex: In circle O, m(AOB = 70(, [pic] and [pic]and [pic] are in the ratio 3:8. Find m(BOC.

Geometry Notes C – 2: Tangents and Chords

Tangents and Chords

A tangent to a circle is a line (in the plane of the circle) that

intersects the circle in

A secant to a circle is a line that intersects the circle in

A chord of a circle is a line segment

Theorem: A tangent and the radius it intersects are

Given: [pic] is tangent to circle O at A; radius [pic] is drawn

Prove:

A tangent segment is a line segment from the point of tangency to another point on the tangent line.

Theorem: Two tangent segments drawn to a circle from the

same external point

Ex: In the diagram at right PA = [pic], PB = [pic] and

the radius of circle O is 2.

a. Find the value of x.

b. Find the length of [pic].

Theorem: If a radius (or diameter) is perpendicular to a chord, then it

Theorem: If two chords of a circle are equidistant from the center,

then they are , and conversely.

Ex: In circle O with radius 12, chord [pic] is 8 units from O.

a. What is the length of the chord?

b. What is the degree measure of [pic]?

Ex: In circle O, [pic] is a tangent segment and [pic].

a. m(AOB =

b. [pic]

c. m(AQO =

d. m(P =

Geometry Notes C – 3: Inscribed Angles

An inscribed angle of a circle is an angle whose vertex is on the circle

and whose sides are contain chords of the circle.

Theorem: The measure of an inscribed angle is

Given: Circle O with inscribed angle (APB.

Prove: [pic]

Case 1: One side of the inscribed angle includes a diameter of the circle.

Case 2: The center of the circle is in the interior of the angle.

Case 2: The center of the circle is in the exterior of the angle.

Corollary: If two inscribed angles intercept the same arc, they are congruent.

Corollary: Congruent inscribed angles intercept congruent arcs and vice versa.

Ex: Find the measure of x in each diagram.

a. b. c. d.

Ex: In circle O, [pic] is a diameter, m(BDC = 30( and

[pic] = 5:4. Find

a. m(CAB

b. m(CBA

c. m(ACD

d. m(CEB

Geometry Notes C – 5: Angles Formed by Chords, Secants and Tangents

“Interior Angle”

An interior angle of a circle is an angle formed by

Theorem: The measure of an interior angle in a circle is half the sum of the

measures of the arcs intersected by the angle and its vertical angle.

“Exterior Angle”

An exterior angle of a circle is an angle formed by

Theorem: The measure of an exterior angle in a circle is half the difference

of the measures of the arcs intersected by the angle.

Ex: Find the measure of x in each diagram.

a. b. c.

d. e. f.

Ex: In the diagram at right, [pic] is a tangent and [pic] = 3:4.

Find

a. m(P

b. m(DEC

c. m(FDA

Geometry Notes C – 7: Chords, Secant/Tangent Segments

Segments Formed by Intersecting Chords

Theorem: When two chords intersect inside a circle,

Ex: Solve for x in the diagram.

Ex: In the diagram (which is not drawn to scale), AE = 12, EB = 6 and

CD = 22. Find the lengths of CE and ED if CE > ED.

Segments Formed by Intersecting Secants (or a Secant and a Tangent)

Vocabulary: [pic] and [pic] are called secant segments.

[pic] and [pic] are called external segments of the secants.

Theorem: When two secants intersect outside a circle, the product of the length of one secant segment and its external segment is equal to the product of the lengths of the other secant segment and its external segment.

Proof:

Ex: Solve for x in the diagram.

Ex: In the diagram (which is not drawn to scale), [pic] is a tangent segment. If PA = [pic] and MC is 4 more than PM, find the length of [pic].

Geometry Notes C – 10: Proofs

The following facts/theorems may be helpful on tonight’s homework and Thursday’s test.

1. All radii of a circle

2. A radius (or diameter) and a tangent to a circle

3. The arcs between two parallel chords are

4a. Inscribed angles that intercept congruent arcs

b. Inscribed angles that intercept the same arc

5. An inscribed angle in a semicircle is

-----------------------

C

r

all a fixed distance r from

a given point C.

congruent

they have the same radius.

A

O

with its vertex at the center of the circle; e.g. (AOB

B

the part of the circle between two points on the circle.

O

B

A

C

D

less than half the circle.

[pic], [pic], [pic] and [pic]. Also [pic].

more than half the circle.

[pic] or [pic]. Named with AT LEAST three letters.

exactly half the circle.

[pic] or [pic].

A

B

Z

Y

O

the same as the measure of

its central angle: [pic]

[pic] and [pic] have the same degree measure but they are NOT the same length.

360(

180(

the same measure. ([pic] and [pic] above are NOT congruent arcs.)

O

B

A

C

D

If (AOB ( (COD, then [pic] and conversely.

[pic] but [pic]

A

C

B

O

P

70(

90(

90(

70( + 90( = 160(

360( - 160( = 200(

3x + 8x + 2(70) = 360

11x + 140 = 360

11x = 220

x = 20

m(BOC = [pic] = 3(20) = 60(

8x

3x

70(

70(

70(

A

C

B

O

D

O

tD

lD

sD

A

B

C

exactly one point. Line t is a tangent.

two points. Line s is a secant.

joining two points on the circle.

[pic] is a chord.

perpendicular.

[pic]

Assume [pic] ( [pic]. Let P be the point on [pic] such that

[pic]. Then OP < OA (leg shorter than hypotenuse in rt [pic]). But that would make point P INSIDE the circle which is impossible if P is on tangent [pic]. Thus, assumption is contradicted and [pic] ( [pic].

O

A

T

N

are congruent. In diagram below, PA = PB.

4

B

P

A

O

2

[pic] ( x(x + 3) = 3(x + 48)

x2 + 3x = 3x + 144

x2 = 144

x = 12 (reject -12)

22 + 42 = (OP)2 ( OP = [pic]

bisects the chord, and conversely.

If [pic], then [pic] bisects [pic] (AM = MB).

If [pic] bisects [pic] (AM = MB), then [pic].

A

M

R

O

B

A

M

C

O

B

D

N

congruent

If OM = ON, then AB = CD.

If AB = CD, then OM = ON.

x2 + 82 = 122

x2 + 64 = 144

x2 = 80

x = [pic] so chord = [pic]

x

8

(

121

[pic]

[pic]

70( (central ( = arc)

O

B

A

P

Q

70

70

x

y

x

180 – 70 = 110(

35(

2x = 70 ( x = 35

20(

70 + 90 + y = 180 ( y = 20

O

A

P

B

one half the measure of the intercepted arc:

[pic] which also means [pic]

O

A

P

B

arc

arc

(

(

Measure of central (AOB = arc

(APO ( (PAO b/c [pic]

So ( + ( = arc ( 2( = arc

[pic]

O

A

P

B

(1

arc1

(2

arc2

From case 1: (1 = [pic]

and (1 = [pic]

So (1 + (1 = [pic]

or ( = [pic]

arc

arc1

O

A

P

B

(2

(1

( = (1 - (2 = [pic]

or ( = [pic]

(

arc2

.

120(

x

.

120(

x

.

70(

x

.

80(

x

Note: Rule does not apply to angle formed by external secant segment and chord.

Note: Rule works for angle formed by tangent and chord.

x = [pic] 70 = [pic]

= 40( x = 2(70)

= 140(

x = [pic] = 60(

180 – 60 = 120

.

O

D

E

30(

C

B

A

2(30) = 60

[pic]

[pic]

[pic]

4x

4(20) = 80

5x

5(20) = 100

[pic]

5x + 4x = 180 ( 9x = 180 ( x = 20

= m(CAE + m(ACE = 50 + 30 = 80(

the intersection (inside the circle) of two chords.

(DEC ( (AEB and (DEA ( (CEB

.

A

D

C

B

E

m(AEB = m(DEC = [pic]

Proof: Draw chord [pic], label angles and arcs as shown.

Then ( = (1 + (2 = [pic]

or ( = [pic]

A

D

C

B

E

arc2

(

(

(1

(2

arc1

the intersection (outside the circle) of two secants, two tangents or one secant and one tangent; e.g. (P

.

A

D

C

B

P

m(P = [pic]

or ( = [pic]

Proof: For HW.

arc1

arc2

(

360 – 170 – 120

= 70

.

120(

x

170(

.

40(

x

130(

.

80(

x

110(

x = [pic] x = [pic] x = [pic]

= 95( = 45( = 50(

40(

.

70(

x

25(

.

70(

x

40(

.

x

360 – x

70 = [pic] 25 = [pic] 40 = [pic]

140 = x + 40 50 = 70 - x 80 = 2x - 360(

x = 100( x = 20( x = 220(

20(

110(

A

C

E

D

B

P

G

F

3x = 3(30)

= 90

= [pic]

= 35(

= [pic]

= 75(

2(20)

= 40

= [pic]

= 45(

4x = 4(3)

= 120

3x + 40 + 4x + 110 = 360 ( x = 30

.

A

D

B

C

E

“Part ( part = part ( part”

One chord Other chord

(DE)(EC) = (BE)(EA)

Proof: HW

.

x

6

8

12

“Part ( part = part ( part”

(8)(12) = 6x

96 = 6x

x = 16

.

D

A

C

B

E

22 - x

“Part ( part = part ( part”

x(22 – x) = (12)(6)

22x – x2 = 72

x2 – 22x + 72 = 0

(x – 4)(x – 18) = 0

x = 4 ( x = 18

So ED = 4 and CE = 18

x

.

A

D

B

C

P

“Outside ( whole = outside ( whole”

One secant Other secant

(PB)(PA) = (PD)(PC)

.

A

D

B

C

P

Draw chords [pic] and [pic].

(PAD ( (PCB b/c they intersect the same arc

(P ( (P by reflexive post.

[pic]PAD ~ [pic]PCB by AA

[pic] b/c corr. sides of ~ [pic]s are in proportion

(PB)(PA) = (PD)(PC) by multiplication post (cross – multiplication)

.

x

6

8

12

“Outside ( whole = outside ( whole”

12(12 + 6) = 8(8 + x)

216 = 64 + 8x

x = 19

.

M

A

C

P

E

[pic]

“Outside ( whole = outside ( whole”

([pic])([pic]) = x(x + x + 4) = x(2x + 4)

96 = 2x2 + 4x

x2 + 2x – 48 = 0

(x + 8)(x – 6) = 0 ( x = 6 (reject x = -8) so PC = 16

x + 4

x

are congruent.

A

B

O

intersect at right angles (are perpendicular).

A

B

C

D

are congruent.

are congruent.

A

C

D

E

B

F

D

A

C

B

a right angle.

A

C

B

O

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