Warm-up 4.2

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

Warm-up 4.2

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

A

, , CA HA GA

, , CG CE DF

DF�������������� �

BJ�������������� �

B

Properties of Tangents

Section 4.2

Standard: MM2G3 ad

Essential Question: How are tangents used to solve problems?

Recall: 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

676 = 676

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

RS

ST R

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

C

50

r70

r

r2 + 702 = (r + 50)2

r2 + 4900 = r2 + 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 ABD DEF, the symbol means “is similar to.”

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 iff to radius and are rt sABE ACD

ABE ACD Reflexive Property

ABE ACD AB AE

AC AD

A

BC D

E

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 iff to radius and are rt sMST PTS

MST PTS Vertical Angles

MNS PNT MN SN

PN TN

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 8 2.8

1 1

5 3m

2

2

1

radius and tangent are

perpendicular

1tm

t