Secants and Tangents Section 10.4

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Secants and Tangents Section 10.4. By: Matt Lewis. Secants and Tangents. -Objectives Identify secant and tangent lines and segments. Distinguish between two types of tangent circles. Recognize common internal and common external tangents . Definitions.

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## Secants and Tangents Section 10.4

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### Secants and TangentsSection 10.4

By: Matt Lewis

Secants and Tangents

-Objectives

• Identify secant and tangent lines and segments.
• Distinguish between two types of tangent circles.
• Recognize common internal and common external tangents.
Definitions
• Secant- a line that intersects a circle a exactly two points. Every secant contains a chord of the circle.
• Tangent- a line that intersects a circle at exactly one point. This point of contact is called the point of tangency.

P

.

M

.

Secant

.

J

Tangent

Definitions Con’t
• Tangent Segment- Part of a tangent line between the point of contact and a point outside the circle.
• Secant Segment- Part a secant line that joins a point outside the circle to the farther intersection point of the secant and the circle.
• External Part of a secant segment- the part of a secant line that joins the outside point to the nearer intersection point.

Tangent Segment

.

.

L

M

.

T

.

A

.

Secant Segment

External Part

M

.

Definitions Con’t
• Tangent Circles- circles that intersect each other at exactly one point.
• Externally Tangent Circles- each of the tangent circles lies outside the other.
• Internally Tangent Circles- one of the tangent circles lies inside the other.

M

.

A

T

- - - - - - - - - -

- - - - - - - -

M

A

T

Definitions Con’t
• Common Tangent- a line tangent to two circles.
• Common Internal Tangent- the tangent lies between the circles. ( WI )
• Common External Tangent- the tangent is not between the circles. ( LM )

I

S

E

W

M

L

Postulates & Theorems
• Postulates
• 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.
• Theorems
• If two tangent segments are drawn to a circle from an exterior point, then those segments are congruent.
Common Tangent Procedure
• Draw the segment joining the centers.
• Draw the radii to the points of contact.
• Through the center of the smaller circle, draw a line parallel to the common tangent.
• Observe that this line will intersect the radius of the larger circle (extended if necessary) to form a rectangle and a right triangle.
• Use the Pythagorean Theorem and properties of a rectangle.
Sample Problems

Sample Problem #1

Step 1 - Constructing radius PB at the point of tangency as shown.

Since lengths of all the radii of a circle are equal, PB = 8.

Step 2 - Since the tangent and the radius at the point of tangency are always perpendicular, ΔABP is a right angled triangle.

Step 3 - Using the Pythagorean theorem,

Step 4 - Substituting for AP, AB and BP,

Step 5 - Since the negative value of the square root will yield a negative value for x, taking the positive square root of both sides,

x = 9.

Given: AC is Tangent to circle P

Calculate the value of X.

Sample Problems

Sample Problem #2

Solution:

OA is AP and OB PB.

A

= 90

O

= 90

140

90 + 90 + 140 + X = 360

X = 40

P

B

PA and PB are Tangents to Circle O.

Find:

Practice Problems

#1

#2

Find:

a, b, and c.

JK is tangent to circles Q & P.

Find: JK

Practice Problems

#3

Given: Two tangent circles, is a common external tangent,

is the common internal tangent.

Prove: D is the midpt. of BC.

Practice Problems

#4

R

P

S

OS = 20

Q

PS = 12

O

What is QS?

#3

• #1- JK = 20.
• #2- = 65

= 25

= 65

• #4- QS = 4

Statements

Reasons

• Two circles are
• externally tangent

1. Given

2. BC is a common

external tangent.

2. Given

3. DA is a common

internal tangent.

3. Given

4. Any two tangents

drawn to a circle from

the same point are .

4. DB DA

5. DC DA

5. Same as 4.

6. DB DC

6. Transitive

7. If a point divides

a line into two seg.,

then it is the midpt.

7. D is the midpt. of

BC.

Practice Exercises
• Pg. 463-464 #1,2,5, & 6.
• Pg. 464-465 #9,10,11-14,16-18.
• These exercises come from our book.
Works Citied

Rhoad, Richard. Geometry for Enjoyment and Challenge. Boston: McDougal Littell, 1991.

Wolf, Ira. Barron’s SAT Subject Test- Math Level I. Barron Publishing, 2008.

Shapes-Circles. http://www.bbc.co.uk/schools/.html.

27 May 2008.

Practice Problems Geometry.

http://www.hotmath.com, 27 May 2008.