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USE PROPERTIES

OF TANGENTS

“THERE ARE NO SECRETS TO SUCCESS. IT IS

THE RESULT OF PREPARATION, HARD WORK,

AND LEARNING FROM FAILURE.” –COLIN

POWELL

CONCEPT 1: PARTS OF

A CIRCLE

Center-point equidistant from the edge of the circle.

Radius-segment from the center to the edge of the

circle.

Diameter-chord that passes through the center of the

circle.

radius diameter

center

CONCEPT 1: PARTS OF

A CIRCLE

Tangent-line that intersects the circle at only one point

called the point of tangency.

Secant-line that intersects the circle twice.

Chord-segment that connects two edges of the circle.

radius

Chord

center

Tangent

EXAMPLE 1

Match the notation with the term that best describes it.

1. 𝑨𝑩

2. 𝑩𝑯

3. 𝑫𝑩

4. 𝑬𝑮

5. Name the circle with notation.

A

B

C

D

E

F

G

H

CONCEPT 2: INTERSECTION

OF CIRCLES

Two circles can intersect in five main ways shown below.

Concentric circles are circles with the same center.

CONCEPT 3: COMMON

TANGENTS

A common tangent is a tangent line, ray or segment that is

tangent to multiple circles.

EXAMPLE 2

Draw the common tangents for circles that intersect only

once (tangent circles).

CONCEPT 4:

THEOREM 10.1

In a plane, a line is tangent to a circle iff the line is

perpendicular to a radius of the circle at its endpoint on the

circle (point of tangency).

radius

center

Tangent

EXAMPLE 3

Determine if 𝑨𝑩 is a tangent line. Explain how you know.

A

C

B

15

BC=17 8

EXAMPLE 4

Find the value of the radius. 𝑨𝑩 is a tangent.

A

C

B

48

36 r

r

CONCEPT 4:

THEOREM 10.2

Tangent segments form a common external point are

congruent.

Ex: If segment AC and segment BC are tangents, then

AC=BC.

B

A C

EXAMPLE 4

Find the value of x.

B

A C 𝑥2 − 4𝑥 + 2

𝑥2 − 4𝑥 − 2

“A DREAM DOESN’T BECOME REALITY THROUGH

MAGIC; IT TAKES SWEAT, DETERMINATION AND

HARD WORK.” –COLIN POWELL

FIND ARC

MEASURES

CONCEPT 5: ARCS AND

NOTATION

An arc of a circle is a portion of the circle. A minor arc

accounts for less than half the circle. A major arc

accounts for more than half the circle. A semicircle

accounts for exactly half of a circle.

Minor arc AB: 𝑨𝑩

Major arc from A through C to B: 𝑨𝑪𝑩

Semicircle from A through B to C: 𝑨𝑩𝑪

B A

C

CONCEPT 5: ARCS AND THEIR

MEASURE

The central angle is the angle formed by the endpoints of an

arc and the center of the circle. The measure of the arc is

equal to the measure of this angle.

m𝑨𝑩 =?

m𝑨𝑩𝑪 =?

m𝑨𝑪𝑩 =?

B A

C

80° 100°

D

Circle D

EXAMPLE 1

CONCEPT 6: ARC

ADDITION POSTULATE

The measure of an arc formed by two adjacent arcs is the

sum of the measures of the two arcs.

Ex: m𝑨𝑩 +m𝑩𝑪 =m𝑨𝑩𝑪

B A

C

80° 100°

D

Circle D

EXAMPLE 2

If the measure of arc AB is 79o and the measure of arc BC is

115o, what is the measure of arc ABC?

EXAMPLE 3

144°

36°

108°

72°

Missing Assignments

1st Qtr

2nd Qtr

3rd Qtr

4th Qtr

A

B

C

D

Find the measure of these arcs:

1. m𝐴𝐵𝐶

2. m𝐴𝐵𝐷

3. m𝐷𝐵𝐶

4. m𝐷𝐴𝐵

5. m𝐶𝐷𝐴

CONCEPT 7: CONGRUENT

CIRCLES AND ARCS

Circles are congruent if they have the same radius.

(consequently if they have the same diameter)

Arcs are congruent if they have the same measure and are of

the same circle or congruent circles.

EXAMPLE 4

APPLY

PROPERTIES

OF CHORDS

“INDIVIDUAL COMMITMENT TO A GROUP EFFORT –

THAT IS WHAT MAKES A TEAM WORK, A COMPANY

WORK, A SOCIETY WORK, A CIVILIZATION WORK.” –

VINCE LOMBARDI

CONCEPT 8: THEOREM 10.3

In the same circle or in congruent circles, two minor arcs are

congruent iff their corresponding chords are congruent.

EXAMPLE 1

Find the measure of the arc given information provided

below.

RS=ST and m𝑹𝑻 =70°

Find m𝑹𝑺 .

EXAMPLE 2

Find the measure of the arc given information provided

below.

RS=ST and m𝑺𝑻 =70°

Find m𝑹𝑺 .

CONCEPT 9: THEOREM 10.4

If one chord is a perpendicular bisector of another chord,

then the first chord is a diameter.

diameter

EXAMPLE 3

CONCEPT 9: THEOREM 10.5

If a diameter of a circle is perpendicular to a chord, then the

diameter bisects the chord and its arc.

EXAMPLE 4

P

P

P

CONCEPT 10: THEOREM

10.6

In the same circle, or in congruent circles, two chords are

congruent iff they are equidistant from the center.

EXAMPLE 5

Find QR, QU, and the radius of circle C if ST=32, UC =12, and

CV=12.

USE INSCRIBED

ANGLES AND

POLYGONS

“CHOOSE A JOB YOU LOVE, AND YOU WILL

NEVER HAVE TO WORK A DAY IN YOUR

LIFE.” –CONFUCIUS

CONCEPT 11: MEASURE OF

INSCRIBED ANGLE

An inscribed angle is an angle whose vertex is on the edge of

the circle and its sides are chords.

Intercepted arc is the arc formed by an inscribed angle.

The measure of an inscribed angle is one half the measure of

the intercepted arc.

Inscribed angle

A

B

Intercepted arc

EXAMPLE 1

CONCEPT 12:

THEOREM 10.8

If two inscribed angles of a circle intercept the same arc, then

the angles are congruent.

CONCEPT 13: INSCRIBED

POLYGONS

An inscribed polygon has all of vertices on the edge of

another polygon/shape.

A circumscribed circle is a circle that contains a

polygon/shape on the inside.

CONCEPT 14:

THEOREM 10.9

Part 1: If a right triangle is inscribed in a circle, then the

hypotenuse will be a diameter of the circle.

Part 2: If one side of an inscribed triangle is the diameter of

the circle, then the triangle is a right triangle and the angle

opposite the diameter is the right angle.

diameter

EXAMPLE 3

Find the value of each variable.

CONCEPT 14:

THEOREM 10.10

A quadrilateral can be inscribed in a circle iff its opposite

angles are supplementary.

A

B

C

D

EXAMPLE 4

Find the value of each variable.

EXAMPLE 5

Which shapes can ALWAYS be inscribed in a circle? Explain.

1) Rectangle

2) Right Triangle

3) Isosceles Triangle

4) Scalene Triangle

5) Parallelogram

6) Kite

7) Trapezoid

8) Isosceles Trapezoid

APPLY OTHER

ANGLE

RELATIONSHIPS

IN CIRCLES

“OPPORTUNITY IS MISSED BY MOST PEOPLE BECAUSE IT

IS DRESSED IN OVERALLS AND LOOKS LIKE WORK.”

–THOMAS A. EDISON

LINES INTERSECTING

CONCEPT 15: THEOREM

10.11

If a tangent and a chord intersect at a point on a circle, then

the measure of each angle formed is one half the measure of

its intercepted arc.

EXAMPLE 1

CONCEPT 16: ANGLES INSIDE

OF THE CIRCLE

If two chords (or two secants) intersect inside a circle, then

the measure of each angle is one half the sum of the measure

of the arcs intercepted by the angle and its vertical angle.

EXAMPLE 2

Find the value of z.

CONCEPT 17: ANGLES OUTSIDE

THE CIRCLE

If a tangent and a secant, two tangents, or two secants

intersect outside a circle, then the measure of the angle

formed is one half the difference of the measures of the

intercepted arcs.

EXAMPLE 3

Find the value of the variable.

EXAMPLE 4

FIND SEGMENT

LENGTHS IN

CIRCLES

“THE BEST PREPARATION FOR GOOD

WORK TOMORROW IS TO DO GOOD WORK

TODAY.” –ELBERT HUBBARD

CONCEPT 18: SEGMENTS OF

CHORDS

Segments of chords are smaller segments that make up a

chord.

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.

Ex: bd=ac

EXAMPLE 1

Find the value of x.

CONCEPT 19: SEGMENTS OF

SECANTS

Secant segment is a segment that contains a chord of a

circle and has only one endpoint outside the circle.

The external segment of a secant segment is the portion of

the secant segment outside the circle.

Theorem: If two secant segments share the same endpoint

outside a circle, then the product of the lengths of one secant

segment and its external segment equals the product of the

lengths of the other secant segment and its external

segment.

Ex: b(b+a)=c(c+d)

EXAMPLE 2

Find the value of x.

EXAMPLE 3

CONCEPT 20: SEGMENTS OF

TANGENTS AND SECANTS

If a secant segment and a tangent segment share an

endpoint outside a circle, then the product of the lengths of

the secant segment and its external segment equals the

square of the length of the tangent segment.

Ex: y(y+x)=z(z)

y(y+x)=𝒛𝟐

EXAMPLE 4

Find the value of x.

EXAMPLE 5

WRITE AND

GRAPH

EQUATIONS OF

CIRCLES

“THE ONLY PLACE SUCCESS COMES

BEFORE WORK IS IN THE DICTIONARY.” –

VINCE LOMBARDI

CONCEPT 21: STANDARD

EQUATION OF A CIRCLE

The standard equation of a circle with center (h, k) and radius

r is: (𝒙 − 𝒉)𝟐+(𝒚 − 𝒌)𝟐= 𝒓𝟐.

EXAMPLE 1

Write the standard equation of the circle.

1. Center (-5, 2) and radius 3

2. Center (2, -4) and radius 4

3. Center (0, 0) and radius 1

CONCEPT 22: GRAPH

EQUATIONS OF A CIRCLE

To graph an equation of a circle:

1. Figure out the center and radius of the circle

2. Plot the center of the circle in the graph

3. Measure the radius right, left, up, and down.

1. Create 4 new points with those measures

4. Connect the four dots with arcs.

EXAMPLE 2

Graph these equations of a circle.

CONCEPT 23: WRITE

EQUATION FROM A GRAPH

To write an equation from a graph:

1. Identify the center and radius of the circle.

2. Plug the center and radius into the standard equation of a

circle.

3. Square r.

EXAMPLE 3

Write the equation for each circle.