Warm Up Section 3.1

Draw and label each of the following:

1. 2. 3.

AB CD GH

4. 5. A point , N, in the exterior of

KLPKLP

6. A point, W, in the interior of KLP

Warm Up Section 3.1

Draw and label each of the following:

1. 2. 3.

AB CD GH

4.

5. A point , N, in the exterior of

KLP

KLP

6. A point, W, in the interior of KLP

AB

C

DG

H

K

L

P

NW

Intro to Circles and Properties of Tangents

Section 3.1

Standard: MCC9-12.G.C.1,2,5

Essential Question: How are tangents used to solve problems?

Introduction: A circle is the set of all points in a plane at a given distance from a given point.

A. A circle is named by its center. The circle shown below has center C so it is called circle C. This is symbolized by writing C.

C

D

A

F

E

B

B. Draw a line segment by connecting points C and D in the circle above. The segment you have drawn is called a radius. The plural of radius is radii. Name three other radii of the circle. Be sure to use the correct notation for a line segment: ______ , ______ , ______Note: one endpoint of the radius is the center of the circle and the other endpoint is a point on the circle. Also, all radii of a circle are congruent. So, if AC = 2 cm, then CE = _____ cm.

C

D

A

F

E

B

CA CE CB

2

C. Now, make another segment by connecting points A and E. The segment you have drawn is called a chord. Name five other chords of the circle. ______ , ______ , ______ , ______ , ______

Note: both endpoints of a chord are points on the circle. Chords of a circle do not necessarily have the same length.

C

D

A

F

E

B

DA DBDE BE BA

D. If a chord passes through the center of a circle it is given a special name. It is called a diameter. Name the diameter pictured in C. _________ Now, draw that diameter.

Note: a diameter is the longest chord of a circle. Its length is twice that of the radius. So, if AC = 2 cm, then AB = _____ cm.

C

D

A

F

E

B

BA

4

E. Next, draw a line passing through points B and F. The line you have drawn is called a tangent. A tangent lies in the plane of the circle and intersects the circle in only one point. The point of intersection is called the point of tangency. What is the point of tangency for BF? ________

C

D

A

F

E

B

B

F. Now, draw the line passing through points A and F. This line is called a secant. Any line that contains a chord of a circle is called a secant.

C

D

A

F

E

B

Use the circle below to identify the following:Name the circle ________

Name all radii pictured. _____________ Name all of chords pictured. __________ Name a diameter. ________ Name a tangent. ________ Name a secant ________

Y

K

WZ

X

XK YK ZK

YZ YX ZX

ZX

ZW

YZ�������������� �

YX�������������� �

ZX�������������� �

K

The points located inside a circle are called interior points.The points located on the circle are points of the circle itself.The points located outside the circle are called exterior points.  Draw four lines that are tangent to both of the circles below at the same time. These lines are called common tangents. 

The two common tangents that pass between the two circles above are called internal common tangents.

The other two tangents are called external common tangents.

Draw all the common tangents possible for the problems below. 1. 2.

3.     

Summary: Circle: The set of all points in a plane at a given distance from a given point is a circle. P is the set of all points in the plane that are 2 units from P. The given point P is the center of the circle.

 Radius: The given distance is the radius of the circle. A radius is also a segment joining the center of the circle to a point of the circle. (The plural of radius is radii.)  Chord: A segment whose endpoints lie on a circle is a chord.  Diameter: A chord that contains the center of a circle is a diameter. A diameter is also the length equal to twice a radius.  

P

Secant: A line that contains a chord of a circle is a secant. Tangent: A line in the plane of a circle that intersects the circle in exactly one point is a tangent. The point of tangency is the point of intersection.

radiusdia

met

er

seca

nt

tangent

chord

Identify each of the following from the diagram below.

1. Center

2. 3 radii

3. 3 chords

4. Secant

5. Tangent

6. Point of Tangency

CA

B

D E

G

H

FJ

Sticky Note Problem

Identify each of the following from the diagram below.

1. Center

2. 3 radii

3. 3 chords

4. Secant

5. Tangent

6. Point of Tangency

C

A

B

D E

G

H

FJ

A

, , CA HA GA

, , CG CE DF

DF�������������� �

BJ�������������� �

B

Sticky Note Problem

REMEMBER: a tangent is a line in the plane of a circle that intersects the circle in exactly one point, the point of tangency. A tangent ray and a tangent segment are also called tangents.

Theorem 1: In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle (the point of tangency).

For the figure at right, identify the center of the circle as O and the point of tangency as P. Mark a square corner to indicate that the tangent line is perpendicular to the radius.

O

P

Theorem 2 : Tangent segments from a common external point are congruent. Measure and with a straightedge to the nearest tenth of a cm.RS = _______ cm RT = ______ cm

T

S

R

2.6 cm

2.6 cm

2.6 2.6

RS RT

Example 1: In the diagram below, is a radius of circle R. If TR = 26 , is tangent to circle R?

S

R

ST

T1024

26

Right Triangle?102 + 242 = 262

100 + 576 = 676676 = 676

Therefore, ∆RST is a right triangle.So, is tangent to R.

RS

ST

Example 2: is tangent to C at R and is tangent to C at S. Find the value of x.

QR

S

R

Q

32

3x + 5

32 = 3x + 527 = 3x 9 = x

QS

RQ SQ

Example 3: Find the value(s) of x:

S

R Q

x2

16

x2 = 16 x = ±4

Example 4: In the diagram, B is a point of tangency. Find the length of the radius, r, of C.

B

C50

r70

r

r2 + 702 = (r + 50)2

r2 + 702 = (r + 50)(r + 50) r2 + 702 = r2 + 50r + 50r + 2500r2 + 4900 = r2 + 100r + 2500 4900 = 100r + 2500 2400 = 100r

24 = r

Recall: Two polygons are similar polygons if corresponding angles are congruent and corresponding sides are proportional. In the statement ABC DEF, the symbol means “is similar to.” 

A

B

CD

E

F

Triangle Similarity Postulates and Theorems: Angle-Angle (AA) Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. Side-Side-Side (SSS) Similarity Theorem: If the corresponding side lengths of two triangles are proportional, then the triangles are similar. Side-Angle-Side (SAS) Similarity Theorem: If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

Example 5: In the diagram, the circles are concentric with center A. is tangent to the inner circle at B and is tangent to the outer circle at C. Use similar triangles to

show that .

BE CD

AB AE

AC AD

A

BC D

E

1. 1. ________________  2. _____________________ 2. Definition of  3. _____________________ 3. All right angles are  4. CAD BAE 4. _________________ 5. _____________________ 5. AA Similarity Postulate 6. _____________________ 6. Corresponding lengths

of similar triangles are in proportion

and AB BE AC CD Tangent if to radius and are rt sABE ACD

ABE ACD Reflexive Property

AB AE

AC AD

A

BC D

E

ABE ACD

Example 6: In the diagram, is a common internal tangent to M and P. Use similar triangles to show that

ST

MN SN

PN TN

M

N T

PS

1. 1. ________________2. _____________________ 2. Definition of  3. _____________________ 3. All right angles are  4. MNS PNT 4. ________________ 5. _____________________ 5. AA Similarity Postulate 6. _____________________ 6. Corresponding lengths of

similar triangles are in proportion

M

N T

PS

and MS ST PT ST Tangent if to radius and are rt sMST PTS

MST PTS Vertical Angles

MN SN

PN TN

MNS PNT

Radius and tangent are perpendicular, hence their slopes are opposite reciprocals

Example 7: Use the diagram at right to find each of the following: 1. Find the length of the radius of A. 

2. Find the slope of the tangent line, t.

A (3, 1)

(5, -1)

2 25 3 1 1 4 4

1

1tm

t

22

35

11

rm

D =

m =

≈ 2.8