TOPIC VIICIRCLES

VII: CirclesESSENTIAL CONTENT

1. Tangent Properties a) Tangent to a Circle b) Tangent Segments

2. Chord Properties a) Chord Central Angles b) Arcs c) Perpendicular to a Chord d) Chord Distance to Center e) Perpendicular Bisector of a Chord

3. Arc and Angle Properties a) Inscribed Angles b) Inscribed Angles Intercepting Arcs c) Angles Inscribed in a Semicircle d) Cyclic Quadrilateral e) Parallel Lines Intercepted Arcs

4. Circumference/Diameter Ratio5. Arc Length 6. Applications in Mathematics and the Real-World

Lesson 6.1Tangent Properties

TOPIC VII - CIRCLES

TANGENT PROPERTIES

Relationship between a tangent line to a circle and the radius of the circle

Relationship between two tangent segments to a common point outside the circle

You will learn:

TANGENT PROPERTIESTangent DefinitionA line that intersects a circle at exactly one point called point of tangency

B

F

E

Example: EF

TANGENT PROPERTIESTangent conjecture

A tangent to a circle is perpendicular to the radius draw to the point of tangency

O

T

TANGENT PROPERTIESTangent Segments Conjecture

Tangent segments to a circle from a point outside the circle are congruent

N

E

A

G

AN GN

TANGENT PROPERTIESTangent Segments conjecture

O

A

B

400 The measure of a minor arc is defined as the measure of its central angle, so AB = 400

The measure of a major arc is the reflex measure of BOA

or 3600 - the measure of the minor arc

BOA determines the minor arc, AB

BOA is said to intercept AB because is within the angle.

m BCA = 320

TANGENT PROPERTIESTangent Segments conjectures

In the figure at right TA and TG are both tangent to the circle N. If the major arc formed by the two tangents measure 2200, find the measure of T

The minor arc intercepted by N measures 3600 – 2200 or 1400

Solution:

TN

A

G

2200

Example:

m N = 1400

A and G must be right angles By quadrilateral conjecture the sum of the angles in TANG is 3600

So, m T + 90 0 + 140 0 + 90 0 = 360 0 means that m T = 40

TANGENT PROPERTIESTangent circles:Are two circles that are tantgent to the same line at the same point. They can be:

Internally tangent Externally tangent

TANGENT PROPERTIES

The main points of this lesson are that Every line tangent to a circle is perpendicular to the radius at the point of tangency The two tangent segments from a point outside the circle are congruent.

Summary

TANGENT PROPERTIESTangent Segments conjectures

1. Rays m and n are tangent to circle P. Find mw

Pw

1300

m

n

2. Rays r and s are tangent to circle Q. Find x

Q

x

700

r

n

Exercises:

TANGENT PROPERTIESTangent Segments conjectures

3. Ray K is tangent to circle R. Find y

RK

1200

y

4. Line t is tangent to both tangent circles. Find z

M

750

r

Exercises:

z

t

S