**Circles**

**G.11 G.12**

A

circle

is the set of all of the points that are equidistant from a given point in a plane

The

radius

is the distance from the center to a point on the circle

A

chord

is a segment joining two points on the circle.

A

diameter

is a chord that passes through the circle’s center.

A

secant

is a line that contains a chord. It intersects the circle at two points

A

tangent

is a line that intersects a circle at only one point.

An

arc

is an unbroken part of a curve of a circle.

The

arc measure

is the degree measure of its central angle.

A

central angle

is an angle with its vertex at the circle’s center.

A

major arc

measures greater than 180º but less than 360º

A

minor arc

measures greater than 0º but less than 180º

Semicircles

are the two arcs of a circle that are cut off by a diameter. A semicircle measures 180

**Vocabulary**

The

intercepted arc

is the part of the circle that lies between the two lines that intersect the circle.

An

inscribed angle

is an angle whose vertex is on the circle and whose sides are chords of the circle.

The measure of an inscribed angle is equal to one-half the measure of its intercepted arc.

The length of an arc (

arc length

) is a linear measure

and is part of the

circumference

(perimeter of a circle).

A

sector

of a circle is that part of the circle

bounded by two radii and an arc

Central and Inscribed Angles

A

central angle

is equal to the

measure of the intersected arc

An

inscribed angle

is equal to half the measure of the intersected arc or half the measure of the

central angle with the same endpoints

Try It!

Find the measure of angle

PQR

Try It!

Find the measure of angle

PQR

120 = 2(PQR)

60 = PQR

Now go to the class

page and take the

central and inscribed

angles mini quiz

Area and Circumference

Formulas

We can use proportions

to find the arc length and

sector area of a given part

of a circle

Arc Measure / Angle

360

Sector Area / Arc Length

Circle Area / Circumference

=

Try It!

Find the arc length and area of the yellow sector

10

30

o

Try It!

Find the arc length and area of the yellow sector

10

30

o

Circumference = 2(10) = 20

Area = 10 = 100

2

Arc Length

30 x

= =

360 20

600 = 360x

x =

5

3

Arc Length

300 = 360y

30 y

= =

360 100

y =

5

6

Now go to the

school fusion page

and complete the

mini quiz on

area and circumference

Angle Relationships

When two chords intersect inside

a circle, the product of the lengths

of the segments of one chord equals

the product of the lengths of the

segments of the other chord.

The measure of an angle formed

by two chords that intersect inside

a circle is equal to half the sum of

the measures of the intercepted arcs.

Two Chords

Interior Angles

Try It!

If BP = 8, PC =6 , AP = 4,

PD = 2x - 8, find the value

of x.

Try It!

If BP = 8, PC =6 , AP = 4,

PD = 2x - 8, find the value

of x.

(BP)(PC) = (AP)(PD)

(8)(6) = (4)(2x - 8)

48 = 4(2x - 8)

12 = 2x - 8

20 = 2x

10 = x

Try It!

Find the

value of x

Try It!

Find the

value of x

x = 75 + 65

2

x = 140

2

x = 70

When segments intersect inside a circle, they also form special linear relationships

Two Secants

When two secant segments are drawn to a circle from an exterior point, the product of the lengths of one secant segment and its exterior segment is equal to the product of the lengths of the other secant segment and its exterior segment.

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

Concentric Circles

Two circles that share the same center

are called

concentric circles

We can use the properties of circles

to solve problems about concentric

circles

Ex. Find the area between the two circles

7

3

Area between = Area of Larger Circle - Area of Smaller Circle

= (7) - (3) = 49 - 9 = 40

2

2

Try It!

Find the area between the two circles

4

5

Try It!

Find the area between the two circles

4

5

(5) - (2) = 25 - 4 = 21

2

2

Linear Relationships

Secant and Tangent

When lines intersect in a circle,

they form special angle relationships

Exterior Angles

An angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle.

The measure of the angle formed is equal to ½ the difference of the intercepted arcs.

Two Secants

Two Tangents

Secant and Tangent

Try It!

Find the value of x

Try It!

Find the value of x

Now go take the angle

relationship mini quiz

Try It!

Find the value of x

Try It!

Find the value of x

(AP)(CP) = (DP)(BP)

(2x+10)(10) = (2x+3+9)(9)

20x + 100 = 18x + 108

2x = 8

x = 4

The square of the length of the tangent equals the product of the length of the secant and its exterior segment.

(UV) = (UY)(UX)

2

Try It!

Find the

value of x

Try It!

Find the

value of x

10 = (3x + 5)(5)

2

100 = 15x + 25

75 = 15x

5 = x

Now go to the class site and take the linear relationships mini quiz

Now go to the class

site and take the

concentric circles mini

quiz

Equations of Circles

When a circle is drawn on a coordinate plane, it is defined by the following equation:

(x - h) + (y - k) = r

Where (h,k) is the center of the circle and

r is the radius of the circle

2

2

2

ex. The circle has a

center at (3,4) and a

radius of 5

Try It!

Write the equation of a circle that has

a center at (-2,4) and point (0,4) is on the

circle.

Try It!

Write the equation of a circle that has

a center at (-2,4) and point (0,4) is on the

circle.

(x - h) + (y - k) = r

(0 - (-2)) + (4 - 4) = r

4 + 0 = r

16 + 0 = r

16 = r

4 = r

Answer: (x + 2) + (y - 4) = 16

2

2

2

2

2

2

2

2

2

2

2

2

2

Now take the equations of

circles mini quiz

Chords and Arcs

Congruent chords create special relationships with the arcs they intersect

Theorem # 1

In a circle, if two chords are congruent then their corresponding minor arcs are congruent.

Try It!

Find the measure

of arc AB

160

40

o

o

Try It!

Find the measure

of arc AB

160

40

o

o

AB = DC

160 + 40 + AB+ DC = 360

200 + 2(AB) = 360

2AB = 160

AB = 80

Theorem # 2

In a circle, two chords are congruent if and only if they are equidistant from the.

Try It!

Given that AD = AE

CB = 2x +1

GF = 4x + 19

Find the length of GF

Try It!

Given that AD = AE

CB = 2x +1

GF = 4x - 19

Find the length of GF

2x + 1 = 4x - 19

2x + 18 = 4x

18 = 2x

9 = x

GF = 4x - 19

GF = 4(9) - 19

GF = 36 - 19

GF = 17

Theorem # 3

In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc.

Try It!

If AP = 12 and arc ACB = 140,

Find the length of AB and

the measure of arc CB

Try It!

If AP = 12 and arc ACB = 140,

Find the length of AB and

the measure of arc CB

OC bisects AB and arc ACB, so

AB = 2(AP) and ACB = 2(CB)

AB = 2(AP)

AB = 2 (12)

AB = 24

ACB = 2(CB)

140 = 2CB

70 = CB

Now go to the class site and take the

chords and arcs mini quiz

Tangents

Theorem # 1

If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

Try It!

If OP = 18 and PT = 24,

what is the length of OT?

Try It!

If OP = 18 and PT = 24,

what is the length of OT?

OT = OP + PT

OT = 18 + 24

OT = 324 + 576

OT = 900

OT = 30

2

2

2

2

2

2

2

2

Theorem #2

Try It!

If PR = 3x + 5

and PQ = 9x - 25,

find the length of PR

Try It!

If PR = 3x + 5

and PQ = 9x - 25,

find the length of PR

PR = PQ

3x + 5 = 9x - 25

30 = 6x

5 = x

PR = 3x + 5

PR = 3(5) + 5

PR = 15 + 5

PR = 20

Now go to the class site

and take the tangents mini quiz