9.1 Introduction to Circles

Some definitions you need• Circle – set of all points in a plane

that are equidistant from a given point called a center of the circle. A circle with center P is called “circle P”, or P.

P

Some definitions you need

• A radius is a segment whose endpoints are the center of the circle and a point on the circle.

• QP, QR, and QS are radii of Q. All radii of a circle are congruent. Ignore the PS

P

Q

R

S

Some definitions you need

• A chord is a segment whose endpoints are points on the circle.

• A secant is a line that contains a chord (or basically it intersects the circle at two points)

• A diameter is a chord that passes through the center of the circle.

Some definitions you need

• A tangent is a line in the plane of a circle that intersects the circle in exactly one point, called the point of tangency.

Sphere

• A sphere with center ) and radius r is the set of all points in space at a distance r from point O. Can you determine which line is the radius? Secant? Tangent? Chord?

Ex. 1: Identifying Special Segments and Lines

Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius of C.

a. AD

b. CD

c. EG

d. HB

J

H

B

A

CD

K

G

E

F

Ex. 1: Identifying Special Segments and Lines

a. AD – Diameter because it contains the center C.

b. CD – radius because C is the center and D is a point on the circle.

c. EG – a tangent because it intersects the circle in one point.

d. HB - is a chord because its endpoints are on the circle.

J

H

B

A

CD

K

G

E

F

First Assignment

• Please complete page 330 #s1-6 and turn in to me

• Then continue notes

9.2 Tangents to CirclesGoal: Apply definitions and theorems about

tangents to figures to calculate segment lengths

Purpose: We are studying this material because tangents are segments that

provide missing information about lengths in circles.

lQ

P

Theorem 9.1

• If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

• If l is tangent to Q at point P, then l

QP.⊥

l

Definition: Common Tangents – a line that is common to two circles

Common internal tangents

Common external tangents

Definition: Tangent Circles – circles that are tangent to the same line at the same

point

These two circles are externally tangent

These two circles are internally tangent

Corollary• If two tangents

originate from the same point, then the tangents are congruent

Ex: Using properties of tangents• AB is tangent to

C at B.• AD is tangent to

C at D.• Find the value of x.

11

AC

B

D

x2 + 2

Solution:11

AC

B

D

x2 + 2

11 = x2 + 2

Two tangent segments from the same point are

Substitute values

AB = AD

9 = x2Subtract 2 from each side.

3 = x Find the square root of 9.

The value of x is 3 or -3.

Ex: Verifying a Tangent to a Circle• You can use the

Converse of the Pythagorean Theorem to tell whether EF is tangent to D.

• Because 112 _ 602 = 612, ∆DEF is a right triangle and DE is perpendicular to EF. So by Theorem 10.2; EF is tangent to D.

60

61

11

D

E

F

Ex. 5: Finding the radius of a circle• You are standing at

C, 8 feet away from a grain silo. The distance from you to a point of tangency is 16 feet. What is the radius of the silo?

• First draw it. Tangent BC is perpendicular to radius AB at B, so ∆ABC is a right triangle; so you can use the Pythagorean theorem to solve.

8 ft.

16 ft.

r

r

A

B

C

Solution: 8 ft.

16 ft.

r

r

A

B

C

(r + 8)2 = r2 + 162

Pythagorean Thm.

Substitute values

c2 = a2 + b2

r 2 + 16r + 64 = r2 + 256 Square of binomial

16r + 64 = 256

16r = 192

r = 12

Subtract r2 from each side.

Subtract 64 from each side.

Divide.

The radius of the silo is 12 feet.

Second Assignment

• Page 335 1-6