11-1 Parts of a Circle

Goal Identify segments and lines related to circles.

Key Vocabulary Circle Center Radius Chord Diameter Secant Tangent Point of Tangency Congruent Circles Concentric Circles

The Concept of Circle The circle is the locus of a moving point which is at constant distance from a fixed point, o. o Locus of point point center

This fixed point (equidistant) inside a circle is called center. Definition: A circle is the locus or set of all points in a plane which are at the same distance (equidistant) from a fixed point inside it. O Center A circle A circle has one and only one center. Symbol: o Name a circle by its center.

A circle divides a plane into three parts. 2. Interior of a circle 3. Exterior of a circle A plane O Center The interior of a circle together with its circumference is called the circular region. 1. The circle

INTERIOR A point is inside a circle if its distance from the center is less than the radius. 

EXTERIOR A point is outside a circle if its distance from the center is greater than the radius. 

A point is on a circle if its distance from the center is equal to the radius. 

Radius Definition: Radius - A line segment that joins any point on the circle to its center. M A point on the circle Center O ⊙

 Radii (plural of radius) of a circle are equal in length.  Infinite number of radius can be drawn in a circle. Radius Center K O L M N

Diameter AB Definition: Diameter - A line segment that joins any two points on the circle and passes through its center. A B A circle O Center ⊙

A circle O M  Infinite number of diameters can be drawn in a circle.  As the radii of a circle are equal in length, its diameters too are equal in length. B Q Center P A N

The length of the diameter (d) of a circle is twice the length of its radius (r). Radius OM Center M O N Radius ON Diameter MN Diameter MN = Radius OM + Radius ON Radius OM = Radius ON

D = ? r = ? D = ?

Definition: Chord - A line segment that joins any two points on the circle. O B A A is a point on the circle B is another point on the circle A line segment that joins point A and B Chord ⊙

Diameter is also a chord of the circle. O Chord CD C D M N K L Chord MN Chord KL Diameter CD

The diameter is the longest chord. O Diameter CD C D M N Chord MN C D M N Chord KL L K

L K M N Chord MN O Center Infinite number of chords can be drawn in a circle. Chord KL Chord GH G H

Example 1: In the following figure identify the chords, radii, and diameters. Chords: Radii: Diameter: O D A B F C E

Name the circle. Answer: The circle has its center at E, so it is named circle E, or ⊙ E. Example 2a:

Answer: Four radii are shown: Name the radii of the circle. Example 2b:

Answer: Four chords are shown: Name a chord of the circle. Example 2c:

Name a diameter of the circle. Answer: are the only chords that go through the center. So, are diameters. Example 2d:

a. Name the circle. b. Name a radius of the circle. c. Name a chord of the circle. d. Name a diameter of the circle. Your Turn: Answer: ⊙ F Answer:

Your Turn: A. Name the circle and identify a radius. A. ⊙ B. ⊙ C. ⊙ D. ⊙

Your Turn: B. Which segment is not a chord? A. B. C. D.

Answer: 9 Formula for radius Substitute and simplify. If ST = 18, find RS. Circle R has diameters and. Example 3a:

Answer: 48 Formula for diameter Substitute and simplify. If RM = 24, find QM. Circle R has diameters. Example 3b:

Answer: So, RP = 2. Since all radii are congruent, RN = RP. If RN = 2, find RP. Circle R has diameters. Example 3c:

a. If BG = 25, find MG. b. If DM = 29, find DN. Circle M has diameters c. If MF = 8.5, find MG. Answer: 58 Answer: 12.5 Answer: 8.5 Your Turn:

A.12 cm B.13 cm C.16 cm D.26 cm If QS = 26 cm, what is the length of RV?

A secant line intersects the circle at exactly TWO points. Secant Line

TANGENT: a LINE that intersects the circle exactly ONE time Tangent Line

Point of Tangency The point at which the tangent line intersects the circle.

RadiusDiameterChordSecant Tangent Centre O

RadiusDiameterChordSecant Tangent Radius OM Centre M O

RadiusDiameterChordSecant Tangent Centre E D Diameter DE O

RadiusDiameterChordSecant Tangent Centre Chord PQ P Q O

RadiusDiameterChordSecant Tangent Centre O A B Secant AB

RadiusDiameterChordSecant Tangent S Centre O D E Tangent DS Point of Tangency E

Name the term that best describes the line. Secant Radius Diameter Chord Tangent

Example 4 SOLUTION a. is a diameter because it passes through the center C and its endpoints are points on the circle. AD b. is a chord because its endpoints are on the circle. HB Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius of C. EG c. b. HB d. JK AD a.

Example 4 c. is a tangent because it intersects the circle in exactly one point. EG d. is a secant because it intersects the circle in two points. JK EG c. b. HB d. JK AD a.

Example 5 Identify a chord, a secant, a tangent, a diameter, two radii, the center, and a point of tangency. C is the center. K is a point of tangency. AB SOLUTION HJ is a chord. is a secant. is a tangent. FG is a diameter. DE is a radius. CE is a radius. DC

No points of intersection (same center) Same center but different radii

Congruent Circles - circles with the same radius. These circles are congruent.

Circles that have congruent radii. 2 2 Circles that lie in the same plane and have the same center. Definitions: Concentric circles : Congruent Circles : A B ⊙ A ≅⊙ B

In a plane, two circles can intersect in two points, one point, or no points. Two Points One Point No Point Coplanar Circles that intersect in one point are called Tangent Circles. Intersecting Circles D. E.

No points of intersection (different center)

1 point of intersection (Tangent Circles) Internally Tangent Externally Tangent

2 points of intersection

Circles in Coordinate Geometry Example 6 Name the coordinates of the center of each circle.a. Name the coordinates of the intersection of the two circles. b. What is the line that is tangent to both circles? Name the coordinates of the point of tangency. c. d.What is the length of the diameter of B ? What is the length of the radius of A ? When a circle lies in a coordinate plane, you can use coordinates to describe particular points of the circle.

Example 6 The intersection of the two circles is the point (4, 0). b. The x- axis is tangent to both circles. The point of tangency is (4, 0). c. SOLUTION a.The center of A is A(4, 4). The center of B is B(4, 2). d.The diameter of B is 4. The radius of A is 4.

Find EZ. The diameters of and are 22 millimeters, 16 millimeters, and 10 millimeters, respectively. Example 7a: 1185

Since the diameter of FZ = 5. Since the diameter of, EF = 22. Segment Addition Postulate Substitution is part of. Simplify. Answer: 27 mm Example 7a: 118 5

Find XF. The diameters of and are 22 millimeters, 16 millimeters, and 10 millimeters, respectively. Example 7b: 1185

Since the diameter of, EF = 22. Answer: 11 mm is part of. Since is a radius of Example 7b: 1185

The diameters of, and are 5 inches, 9 inches, and 18 inches respectively. a. Find AC. b. Find EB. Answer: 6.5 in. Answer: 13.5 in. Your Turn:

Example 8

First, find ZY. WZ + ZY= WY 5 + ZY= 8 ZY= 3 Next, find XY. XZ + ZY= XY = XY 14= XY Since the diameter of is 16 units, WY = 8. Similarly, the diameter of is 22 units, so XZ = 11. WZ is part of radius XZ and part of radius WY. Answer: XY = 14 units

Your Turn: A.3 in. B.5 in. C.7 in. D.9 in.

Assignment Pg #1 – 51 odd