Secants and Tangents Section 10.4

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Secants and TangentsSection 10.4

• By Matt Lewis

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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.

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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
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Definitions Cont

• 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
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Definitions Cont

• 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
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Definitions Cont

• 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
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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.

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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. Observe 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.

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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, ?ABP 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.
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Sample Problems
Sample Problem 2
Solution
OA is AP and OB PB.
A
90
O
90
140
AOBP is a quadrilateral.
90 90 140 X 360
X 40
P
B
PA and PB are Tangents to Circle O.
Find
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Practice Problems
1
2
Find
a, b, and c.
JK is tangent to circles Q P.
Find JK
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Practice Problems
3
Given Two tangent circles, is a common
external tangent, is the common internal tangent.
Prove D is the midpt. of BC.
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Practice Problems
4
R
P
S
OS 20
Q
PS 12
O
What is QS?
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Answers to Practice Problems
3

• 1- JK 20.
• 2- 65
• 25
• 65
• 4- QS 4

Statements
Reasons
1. Given

• Two circles are
• externally tangent

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.
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Practice Exercises

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

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Works Citied

• Rhoad, Richard. Geometry for Enjoyment and
  Challenge. Boston McDougal Littell, 1991.
• Wolf, Ira. Barrons SAT Subject Test- Math Level
  I. Barron Publishing, 2008.
• Shapes-Circles. http//www.bbc.co.uk/schools/.ht
  ml.
• 27 May 2008.
• Practice Problems Geometry.
• http//www.hotmath.com, 27 May 2008.