**Chapter 10: Circles**

**by Lev, Jolie and Sam**

10.2: Congruent Chords

Theorem 77: If two chords are equidistant from the center, then they are congruent.

A

E

B

D

C

X

Y

Theorem 78: If two chords of a circle are congruent, then they are equidistant from the center of the circle.

If AB and CD are equidistant from the center of the circle (XE is congruent to YE), then AB is congruent to CD, and vice versa

10.3: Arcs of a Circle

Types of Arcs

Center

Arc

A

B

Arc -

Two points on a circle and all points on the circle needed to connect the points by a single path

Central angle -

An angle whose vertex is at the center of a circle

minor arc

major arc

Minor arcs are inside central angles, while major arcs are outside central angles.

diameter

Semicircle

Semicircle -

An arc whose endpoints are the endpoints of a diameter

A

B

X

Arcs are labeled with the symbol .

Arcs in this circle include AB (minor) and AXB (major).

The Measure of an Arc

90°

diameter

180°

The measure of a minor arc or a semicircle is the same as the measure of the central angle that intercepts the arc.

115°

245°

The measure of a major arc is 360 minus the measure of the minor arc with the same endpoints.

Congruent Arcs

A

G

H

F

E

C

D

B

Two arcs are congruent whenever they have the same measure and are parts of the same circle or congruent circles.

so...

60°

60°

60°

60°

60°

That means that these two arcs are NOT congruent.

I

J

K

L

Relating Congruent Arcs, Chords and Central Angles

Theorem 79: If two central angles of a circle (or congruent circles) are congruent, then their intercepted arcs are congruent.

Theorem 80: If two arcs of a circle (or congruent circles) are congruent, then the corresponding central angles are congruent.

Theorem 81: If two central angles of a circle (or congruent circles) are congruent, then the corresponding chords are congruent.

Theorem 82: If two chords of a circle (or congruent circles) are congruent, then the corresponding central angles are congruent.

Theorem 83: If two arcs of a circle (or congruent circles) are congruent, then the corresponding chords are congruent.

Theorem 84: If two chords of a circle (or congruent circles) are congruent, then the corresponding arcs are congruent.

10.1: The Circle

A

circle

is the set of all points in a plane that are a given distance from a given point in the plane. The given point is the center of the

circle

, and the given distance is the

radius

. A segment that joins the center to a point on the circle is also called a radius.( The plural of

radius

is

radii

.)

Center

Radius

Circle

Two or more coplanar circles with the same center are called

concentric

circles

.

A point is inside (in the

interior

of) a circle if its distance from the center is less than the radius

A point is outside (in the

exterior

of) a circle if its distance from the center is greater than the radius.

A point is on a circle if its distance from the center is equal to the radius.

.

.

.

.

C

A

O

B

A

chord

of a circle is a segment joining any two points on the circle.

A

diameter

of a circle is a chord that passes through the center of the circle.

.

O

Diameter

Chord

Circumference and Area of a Circle

The area of a circle can be found with the formula

A = π

r

^2

and the circumference (perimeter) of a circle can be found with the formula

C = π

d

where

r

is the circle's radius, d is its diameter, and π ≈ 3.14.

The distance from the center of a circle to a chord is the measure of the perpendicular segment from the center to the chord.

.

O

A

P

B

If a radius is perpendicular to a chord, then it bisects the chord.

.

O

D

A

E

B

If a radius of a circle bisects a chord that is not a diameter, then it is perpendicular to that chord.

.

O

G

F

H

E

The perpendicular bisector of a chord passes through the center of the circle.

.

O

C

Q

P

D

10.6: More Angle-Arc Theorems

If two inscribed or tangent-chord angles intercept congruent arcs, then they are congruent.

An angle inscribed in a semicircle is a right angle.

The sum of the measures of a tangent-tangent angle and its minor arc is 180.

10.4: Secants and Tangents

10.5: Angles Related to a Circle

A Secant is a line that intersects a circle at exactly two points. (Every secant contains a chord of the circle.)

Angles of a Circle

A

B

Secant

.

Inscribed angles

are angles whose vertex are on the circle. The sides are made of two chords.

T

Point of Contact

Tangent-chord angles

are angles whose vertex are on the circle and are made up of a tangent and a chord

A tangent is a line that intersects a circle at exactly one point. This point is called the point of tangency or point of contact.

.

Chord-chord angles

are angles formed by two chords that intersect inside a circle but not at the center

Secant-secant angles

are angles whose vertex is outside the circle and whose sides are made by two secants.

Secant-tangent

angles are angles whose vertex is outside a circle and whose sides are made of a secant and a tangent.

A tangent line is perpendicular to the radius drawn to the point of contact.

If a line is perpendicular to a radius at its outer endpoint, then it is tangent to the circle.

Tangent-tangent angles

are angles whose vertex lie outside a circle and whse sides are made by two tangets

Vertex on circle

A tangent segment is the part of a tangent line between the point of contact and a point outside the circle.

Vertex in circle (not center)

.

Vertex outside circle

P

T

.

Tangent segment

A secant segment is the part of a secant line tht joins a point outside the circle to the farther intersection point of the secant and the circle.

The external part of a secant segment is the part of a secant line that joins the outside point to the nearer intersection point.

.

Q

R

.

Secant-Tangent

External Part

Secant-secant

Tangent-tangent

Chord-chord

Inscribed

Secant Segment

Tangent-chord

If two tangent segments are drawn to a circle from an external point, then those segments are congruent. (Two- Tangent Theorem)

.

Theorem 86: The measure of an inscribed angle or tangent-chord angle (vertex on a circle) is one-half the measure of its intercepted arc

60°

30°

P

X

Y

O

Tangent circles are circles that intersect each other at exactly one point

.

T

.

.

Q

P

Two circles are externally tangent if each of the tangent circles lies outside the other.

T

.

Theorem 87: The measure of a chord-chord angle is one-half the sum of the measures of the arcs intercepted by the chord-chord angle and its vertical angle.

40°

60°

50°

Theorem 88: The measure of a secant-secant angle, a secant-tangent angle, or a tangent-tangent angle (vertex outside a circle) is one-half the difference of the measures of the intercepted arcs.

10°

50°

20°

Two circles are congruent if they have congruent radii.

10.8: The Power Theorems

Theorem 95: If two chords of a circle intersect inside the circle, then the product of the measures of the segments of one chord is equal to the product of the measures of the segments of the other chord. (Chord-chord Power Theorem)

A

B

E

C

D

(AE)(BE) = (CE)(DE)

Q

Theorem 96: If a tangent segment and a secant segment are drawn from an external point to a circle, then the square of the measure of the tangent sgment is equal to the product of the measures of the entire secant segment and its external part. (Tangent-Secant Power Theorem)

T

P

Q

R

(RQ)² = (TP)(PR)

Theorem 97: If two secant segments are drawn from an external point to a circle, then the product of the measures of one secant segment and its external part is equal to the product of the measures of the other secant segment and its external part. (Secant-Secant Power Theorem)

C

D

A

P

B

(PA)(BA) = (CA)(DA)

If two inscribed or tangent-chord angles intercept the same arc, then they are congruent.

10.9: Circumference and Arc Length

is the measure of a circle's perimeter,

found by multiplying pi by the circle's diameter (C = πd).

(but you should already know that.)

is found by this theorem:

Length of PQ= ( )πd

mPQ

360

P

Q

Practice Problems!

A polygon is inscribed in a circle if all of its vertices lie on the circle

10.7 Inscribed and Circumscribd Polygons

.

O

A

B

C

A polygon is circumscribed about a circle if each of its sides is tangent to the circle.

P

Q

R

S

T

.

F

The center of a circle circumscribed about a polygon is the circumcenter of the polygon

R

S

T

U

P

Q

.

F

pg. 598-603 problem 14

R

B

S

Q

A

P

Pg. 598-603 Problem 30

Given: Segment AR is tangent to circle P

Segment RS is a diameter of circle Q

118

Prove: Triangle PAR ~ triangle SBR

°

26

°

Statement

Reason

P

1. Given

1. Given

A

2. Angle ARP is

conruent to angle SRB

D

.

2. Vertical angles are congruent

o

3. Angle RAP is a right

angle

c

3. If a segment is tangent to a circle, it

forms a right angle with the circle

4. Angle RBS is a right

angle

B

4. An angle inscribed in a semicircle is a right angle

5. Angle RBS is

congruent to angle RAP

5. Angle RBS is congruent to angle RAP

6. Triangle PAR ~

triangle SBR

6. ~AA

Given: Diagram as marked, with PA and PD tangent at Circle O.

Find

a: Arc AD

b: Measurement of angle P

Pg. 602, problem 33

1. Angle AFD is 62°

2. Angle AFD and Angle CFB are chord-chord angles.

3. To find arc AD, multiply angle AFD by two and subtract 26° from it. Arc AD is 98°

4. Angle BFD and angle AFC are congruent by vertical angles. Arc BD and Arc AC are both 118°.

5. The mesurement of angle P is 82°

F

find x.

According to the tangent-secant theorem (section 10.8), the square of a tangent is equal to the secant multiplied by its outer part, so 6² = 9x. Therefore, 36 = 9x, and 4 = x.

this is a tangent

pg. 710, problem 43

Prove: If two tangent segments are drawn to a circle from an external point, the triangle formed by these two tangents and any tangent to the minor arc included by them has a perimeter equal to the sum of the measures of the two original tangent segments.

REMEMBER! Two-tangent theorem- If two tangent segments are drawn to a circle from an external point, then those segments are congruent.

**The End**

50°

Tangent

Always look for Waldo!!!

Vertex at the center: Intercepted arc is equal!

Vertex inside (not the center): Half the sum!

Vertex on the circle: Half the arc!

Vertex outside the circle: Half the difference!

**(How many times did you spot Waldo?)**

P

C

D

A

E

B

PC + CD = PA

PE + ED = PB

PC + (CD + ED) + PE = PA + PB

PC + CE + PE = PA + PB

The center of a circle inscribed in a polygon is the incenter of the polygon.

If a quadrilateral is nscribed in a circle, its opposite angles are supplementary.

A

B

C

D

If a parallelogram is inscibed in a circle, it must me a rectangle.

A

B

D

C

Two circles are internally tangent if one of the tangent circles lies inside the other.

A common tangent is a line tangent to two circles. Such a tangent is a common internal tangent if it lies between the circles or a common external tangent if it is not between the circles.

P

Q

X

Y

A

B

Common Tangent Procedure

1. Draw the segment joining the centers

2. Draw the radii to the points of contact.

3. Through the center of the smaller circle, draw a line parallel to the common tangent.

4.Obseve that this line will intersect the radius of the larger circle (extended if necessary) to form a rectangle and a right triangle.

5. Use the Pythagorean Theorem and properties of a rectangle.