Geometry

9.1 An Introduction to Circles

Circle

•The set of all points in a plane at a given distance from a given point.

.P

P is the center of the circle.The circle is the set of all points in the plane of the screen 3 away from P.

3

Radius Plural is radii

•A segment that joins the center of the circle to any point on the circle.

.P

radi

us

Chord

•A segment whose endpoints lie on a circle.

.P

chord

Diameter

•A chord that passes through the center of the circle.

diameter.P

The diameter is _________ as long as the radius.

twice

Secant

•A line that contains a chord of a circle.

SECANT .P

Tangent

•A line in the plane of a circle that intersects the circle in exactly one point.

Tangent .P.Point of tangency

Name:

1) The center:

D

A

F

E

C

B

A

Name:

2) Two diameters:

D

A

F

E

C

B

DBand

FC

Name:

3) A point of tangency:

D

A

F

E

C

B

D

Name:

4) Four radii:

D

A

F

E

C

B

ABand

AFand

AC and AD

Name:

5) A tangent:

D

A

F

E

C

B

ED

Name:

6) A secant:

D

A

F

E

C

B

FC

Name:

7) Six chords:

D

A

F

E

C

B

FBand

DFand

BC

and

DCand

DBand

FC

8) Why is AC not aChord of A:

D

A

F

E

C

B

.

A chord is a segment with both endpoints on the circle.

AC has only one endpoint on the circle.

9) Why is BD not aChord of A:

D

A

F

E

C

B

.

A chord is a segment not a line.

Sphere• Sphere: The set of all points in space given distance

from any given point is a sphere. • Many of the terms used with circles are also used with

spheres.• For example, sphere X has a… center:

radii:

chords: diameter: secants: tangent: point of tangency:

J

L I

K

X

F

G

H

Congruent Circles

congruent circles/spheres Circles (or spheres) are congruent if they have congruent radii.

44

Concentric Circles

concentric circles: Circles that lie in the same plane and have the same center are concentric.

concentric spheres: Concentric spheres have the same center.

Inscribed Polygon• A polygon is inscribed in a circle if each vertex of the

polygon lies on the circle.

• A circle is circumscribed about a polygon if each vertex of the polygon lies on the circle.

The polygon is inscribed in the circle.

Thus, the circle is circumscribed about the polygon.

These two sentences have the same meaning.

In sphere A, draw:

10. a diameter, BC

11. a chord, DE

12. a tangent, CF

13. a secant, DG

A

B C

D E

C

F

D G

• 14. Draw two concentric circles. Draw a tangent to one of the circles. Is it tangent to the other circle?

• 15. Draw a large circle. Inscribe an isosceles triangle in the circle.

• 16. Draw a rectangle. Circumscribe a circle about the rectangle.

No.

Find the value of x.

O

Bx

6

6

120120

12060

30

6

3√3

3

3√33√3

x = 6√3

90

O4

x

x

8

O

60

O

x

1045

4

3x

O

x

Find the value of x. O is the center of each circle.

17. 18.

19. 20.

x = 5

x = 8

x = 4√2

x = 5√2

HW

• P. 330-340 CE 1-11 WE 1-15