10-1 Circles & Circumference

Circle-

Two circles are congruent if and only if they have congruent radii

All circles are similar

Concentric circles are coplanar circles that have the same center

2 points of intersection

1 point of intersection

NO points of intersection

Radius- Formula- Chord- Diameter- Formula-

Circumference-

Formula-

11-3 (part1) Areas of Circles Area of a circle-

Example 4 Find the circumference of a helipad on the top of a hospital if the diameter of the pad is 37 meters.

Example 5 Find the diameter and radius of a circle to the nearest hundredth if the circumference of the circle is 842.5 millimeters.

Example 1 What is the area of the circular putting green shown to the nearest square foot?

Example 2 Use the Area of a Circle to Find a Missing Measure Find the radius of a circle with an area of 46

square centimeters.

11-3 (part2) Sectors Sector of a circle-

Area of a Sector

Example 1) A circular pizza has a diameter of 16 inches and is cut into 10 congruent slices. What is the area of one slice to the nearest hundredth?

Find the area of each shaded sector. Round to the nearest tenth.

ex 2) ex3) ex4) Ex5) Jason wants to make a spinner for a new board game he invented. The

spinner is a circle divided into 8 congruent pieces, what is the area of each piece to the nearest tenth?

10-2 Arcs and Chords TERM: DEFINE: DRAW and LABEL Arc

Central Angle

Minor Arc “shade” minor arc: LY

Semi-Circle “shade” semi-circle: ACB

Major Arc “shade” major arc: LUY Arc Addition Postulate Congruent arcs

1. Using the letters shown in the diagram, name: a. four central angles _____, _____, _____, _____

b. two semicircles _____, _____

c. four minor arcs _____, _____, _____, _____

d. four major arcs _____, _____, _____, _____

2. Using the same circle, find the measure of each arc.

a. _____mBC b. _____mAB c. _____mABC

d. _____mCD e. _____mABD f. _____mBDA

Example 3 Refer to the circle graph. a. Find the measurement of the central

angle for each category. b. Use the categories to identify any arcs that are congruent.

Example 4 Find each measure in F. Example 5 Find the length of ZY . Round to the nearest hundredth.

a. b.

Example 1 Find measures of Central Angles

Find mQAM.

Example 2

In O, mAOB = 35, mDOE = 42, and .

Identify each arc as a major arc, minor arc, or semicircle. Then find its measure.

a. m AB

b. mCDE

c. m AE

OB OC

A

B

F

E

C

D

45

85 X 80

Y

Z

6 in.

X 120

12 cm

Y Z

10-3 Arcs and Chords ARC-CHORD Relationships

Congruent central angles have congruent chords

Congruent chords have congruent arcs

Congruent arcs have congruent central angles

DIAMETER(radius)-CHORD Relationships

In a circle, if a radius (or diam.) is to a chord, then it bisects the chord and its arc

In a circle, the bisector of a chord is a radius (or diam.)

Example 3 Find NP. Example 4 Find QR.

In the same circle or in circles, 2 chords are if and only if they are equidistant from the center

Example 1 In the figures, J K and

MN PQ . Find PQ.

M

N

J

3x + 4

P Q

K

x + 12

Example 2 In S, m PQR = 86. Find m PQ .

P

Q

R

S

8 T

Example 5 In ⨀P, QR = 7x – 20 and TS = 3x. What is x?

10-4 Inscribed Angles

Inscribed Angle: Intercepted Arc: Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc.

Theorem 10.8 Theorem 10.9 An inscribed angle intercepts a semi-circle (or diam.), If a quadrilateral is inscribed in a circle, then if and only if the angle is a right angle. its opposite angles are supplementary.

BC is a diameter. The sum of the angles of a quadrilateral is m BAC D and are supplementary.

mBAC C and are supplementary.

If inscribed angles of a circle intercept the same arc, then the angles are congruent.

A

B C

Quadrilateral ABCD is inscribed in the circle. The circle is circumscribed.

Find the measure of each angle or segment for each figure.

1) mPS 3) mADC 2) m PRU 4) m DAE 5) x= 7) x= 6) m WUT 8) m Y 9) m R 11) x 10) x 12) m B

13) mRUS 14) a 15) m RUS

10-5 Tangents Tangent-

Example 1: Draw the common tangent(s)

Example 3: JK is tangent to H . Find x.

Example 2

1. the center _____

2. diameter _____

3. a point of tangency _____

4. three radii _____, _____, _____

5. a tangent _____

6. two chords _____, _____

Example 4: GH and KH are tangent to F . Find a. Example 5: RS and RT are tangent to Q . Find n.

Common tangent

Thm 10.10: In a plane, a line is tangent to a circle if and only if it is ______________ to a radius drawn to the point of tangency.

Thm 10.11: If 2 segment from the same exterior point are tangent to a circle then they are _____________.

Ex 7: Find x and the perimeter of the triangle. In the picture, the circle is inscribed and the triangle is circumscribed.

a. b.

Example 6:Find x if ZB is tangent to Y .

10-6 Secants, Tangents, and Angle Measures Secant-

Example 1 Find x.

a.

b.

c.

Interior Angles formed by 2 intersecting chords or secants

Formula:

A secant & a tangent intersect at a point of tangency

Formula:

C

A

DT

x

B

135

53

E

R

S

T

x

U

62 144 V

G

K H

L J

x

120

92

Example 2. Find each measure

a. mQPR

R

136

S

Q

P

b.m

F

D

82 C

E

Vertex of Angle PICTURE Angle Measure

On the circle

Inside the circle

Outside the circle

ex5) x= ex6) x= ex7) x=

10-7 Special Segments in a Circle CHORD-CHORD THM Formula:

SECANT-SECANT-THM Formula:

TANGENT-SECANT THM Formula:

Find x in each picture. ex1) ex2) ex3) ex4)

Find x in each picture. ex5) ex6)

Find x in each picture. ex7) ex 8)

ex9) ex10)

ex11) ex12)

10-8 Equations of Circles Equation of a Circle in Standard Form: Example 1 Write the equation of each circle. a. center at (1, -5), d = 6 b. center at (-6, 0), r = 8 Example 2 Write the equation of the circle with center at (–3, 5) that passes through (3, –3).

Example 3 The equation of a circle is x2 + y2 – 2x – 10y = –22. State the coordinates of the center and the measure of the radius. Then graph the equation.

Example 5 – Intersections with Circles

Find the point(s) of intersection between x2 + y2 = 20 and y = –2x.