Name:____________________________ - North … · Web viewLesson 7: Angle Measures and Segment Lengths (PH text 12.4) Lesson 8: Circles in the Coordinate Plane (PH text 12.5) Lesson

Geometry 2205 Mrs. BondiUnit 4:

GeometryUnit 4: Circles

Unit 4 Circles Topics:

Lesson 1: Circles and Arcs (PH text 10.6)Lesson 2: Areas of Circles, Sectors, and Segments of Circles (PH text 10.7)Lesson 3: Geometric Probability (PH text 10.8)Lesson 4: Tangent Lines (PH text 12.1)Lesson 5: Chords and Arcs (PH text 12.2)Lesson 6: Inscribed Angles (PH text 12.3)Lesson 7: Angle Measures and Segment Lengths (PH text 12.4)Lesson 8: Circles in the Coordinate Plane (PH text 12.5)

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Lesson 1: Circles and Arcs (PH text 10.6)Objectives: to find the measures of central angles and arcs of circles.

to find the circumference of a circle and the length of an arc

Circle: the set of all points equidistant from a given point called the ______________

Circles are named using the symbol “ ” and the center point.

Radius:

Diameter:

Congruent Circles:

Central Angle:

The center P of the circle is the midpoint of the diameter! Remember the midpoint formula:

Example 1: A diameter of a circle has endpoints A(-3, -2) and B(1, 4). Find the coordinates of the center and find the length of the radius.

Arc:2


Arc notation:

Three Types of Arcs: Semicircle:

Named by:

Measure:

Minor Arc:

Named by:

Measure:

Major Arc:

Named by:

Measure:

Example:Find each arc in the diagram to the right and its measure.

Adjacent Arcs:

Postulate 10-2 Arc Addition Postulate: The measure of the arc formed by two adjacent arcs is the sum of the measure of the two arcs.

Example 2:

Circumference -

3

Find the measure of each arc: a. b.

c. d.


Theorem 10-9 Circumference of a CircleThe circumference of a circle is times the diameter.

Example 3: The diameter of a circle is 16 cm. Find the circumference. (round to tenth)

Example 4: The radius of a circle is 4 in. Find the circumference. (round to tenth)

Concentric Circles –

Example 5: Compare circumferences

Arc Length –

Theorem 10-10 Arc Length

The length of an arc of a circle is the product of the ratio

and the circumference of the circle.

length of

Example 6:

4

= pi = = ratio of circumference to diameter of a circle


If , diameter = 12, find length .(round to nearest tenth)

Example 7:If radius = 12 cm, = 210, find length .(round to nearest tenth)

Congruent Arcs -

Practice:

HW: p.654 #8-34 even

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Lesson 2: Areas of Circles, Sectors, and Segments of Circles (PH text 10.7)Objective: to find the areas of circles, sectors and segments of circles

Theorem 10-11 Area of a Circle The area of a circle is the product of and the square of the radius.

Example 1: The radius of a circle is 15 cm. Find the area.

Sector of a Circle – region bounded by two radii and their intercepted arc.

Theorem 10-12 Area of a Sector of a Circle

The area of a sector of a circle is the product of the ratio

and the area of the circle.

Area of sector

Example 2: The diameter of a circle is 8.2 m, Example 3:and = 125. Find the area of sector ADB. Find the area of sector GPHRound to the nearest tenth. Leave your answer in terms of .

Segment of a circle – The part of a circle bounded by an arc and the 8


segment joining its endpoints.

Example 4: A circle has radius 8 cm. Find the area of a segment of the circle bounded by a 120 degree arc. Round your answer to the nearest tenth.

Example 5: What is the area of the shaded segment shown at the right? Round your answer to the nearest tenth.

Practice:

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HW: p.663 #6-30 even

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Lesson 3: Geometric Probability (PH text 10.8)Objective: to use segment and area models to find the probability of events

Geometric probability uses geometric figures to model occurrences of real-life events. The occurrences can be compared by comparing the measurements of the figures.

Reminder:Probability is … P(event) = # of favorable outcomes

# of possible outcomes

Examples:1)

2.

3. KYW gives a weather update every 10 minutes. If you turn on the radio at a random time, what is the probability that you would wait more than 4 minutes to hear the weather update?

P(wait > 4 min.) =

P(wait ≤ 2 min.) =

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4. Bill takes the train to work in Center City each morning. Because of traffic, he cannot be sure exactly when he will arrive at the station. If trains leave the station every 20 minutes during rush hour, what is the probability he will not need to wait more than 5 minutes for a train to leave?

Examples:5.

6.

7. What is the probability of a tossed coin landing in the shaded region of the rectangle below?The rectangle measures 125 cm long and 25 cm wide.

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8. Rebekah and Bridget made a dodecagon shaped dartboard for a school carnival. It is divided into a series of triangles by connecting opposite vertices alternately colored red, orange, yellow, green, blue and purple.

a) What is the probability that the dart will land in a yellow triangle?

b) What is the probability that the dart will land in a red or orange triangle?

c) What is the probability that the dart will land in a primary color triangle?

d) What is the probability that the dart will land in a triangle whose color name contains the letter r?

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HW: p.671 #8-40 even

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Lesson 4: Tangent Lines (PH text 12.1)Objective: to use properties of a tangent to a circle

A Tangent to a Circle -

Point of Tangency –

Theorem 12-1 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

If is tangent to at P, then .

Example 1: Example 2:

Example 3:What is the distance to the horizon that a person can see on a clear day from an airplane 2 miles above the earth? The Earth’s radius is about 4000 mi.

Theorem 12-2 If a line in the same plane as a circle is perpendicular to a radius at (Converse of 12-1) its endpoint on the circle, then the line is tangent to the circle.

If at P, then is tangent to .

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Tangent to a circle

Point of Tangency

A

B


Example 4: Example 5:Find the length of the radius if Find the length of the segment connecting the PQ = 8 and TQ = 5. centers of the circles.

Theorem 12-3 If two tangent segments to a circle share a common endpoint outside the circle, then the two segments are congruent.

If and are tangent to at A and C, respectively, then .

The sides of the triangle to the right are tangent to the circle.The circle is “inscribed in” the triangle.

The triangle is “circumscribed about” the circle.

Example 5: is inscribed within ∆PQR. Label each length that you know based on Thm 12-4.

If ∆PQR has a perimeter of 88cm, find QY.

20

8Q

5T

P

S

6 in

20 in

10 in

17 cm

15 cm

O

Z

R

Y

QXP


Practice:

HW: p.766 #5-1921


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Lesson 5: Chords and Arcs (PH text 12.2)Objective: to use congruent chord, arcs and central angles

to use perpendicular bisectors to chords

Chord – a segment with endpoints on a circle– labeled as a line segment,

– marks an arc on the circle, labeled

Theorem 12-4 and its converse In the same circle or in congruent circles,

a) congruent central angles have congruent arcsb) congruent arcs have congruent central angles


a) congruent central angles have congruent chordsb) congruent chords have congruent central angles

Theorem 12-6 and its converseIn the same circle or in congruent circles,

a) congruent chords have congruent arcs b) congruent arcs have congruent chords

Let’s prove Theorem 12-6 part a.

Given: Prove:

Statements Reasons

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Q

P

O

B

C

O P

F

D



c) chords equidistant from the center(s) are congruentd) congruent chords are equidistant from the center(s)

Example 1: Find the value of x in the diagram to the right. Justify your answer.

Theorem 12-8In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc.

Theorem 12-9 In a circle, if a diameter bisects a chord (that is not a diameter), then it is perpendicular to the chord.

Theorem 12-10In a circle, the perpendicular bisector of a chord contains the center of the circle.

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Example 2:

Let chord 30 in. and 6 in. from the center.

a) Find the radius of b) Find

Example 3: Let chord 22 in. and 7 in. from the center.

a) Find the radius of

b) Find the length of

Practice:

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30 in

6 in

A

TEC

C ET

A

A BD

C

C

DBA


3. What is the missing length to the nearest tenth?

HW: p.776 #5-15, 30-3227


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Lesson 6: Inscribed Angles (PH text 12.3)Objective: to find the measure of an inscribed angle

to find the measure of an angle formed by a tangent and a chord

Theorem 12-10 Inscribed Angle TheoremThe measure of an inscribed angle is half the measure of its intercepted arc.

Example 1:

Find a and b

30

O

C

B

A

70

b

R

S

25T

aP

Q


Example 2:

1) Name a pair of congruent inscribed angles

2) Name a right angle

3) Name a pair of supplementary angles

Example 3:

Find the value of each variable.

Example 4:

Find the value of each variable.

31

F

E D

C

BA

z

yx 98

107


Theorem 12-12The measure of an angle formed by a chord and a tangent (that intersect on a circle) is half the measure of the intercepted arc.

Example 5: Find the value of each variable.

Practice:

HW: p.784 #6-18, 20-21, 23-25

32

A

B58

y

x

zC

D


Extra Practice:

Good for Review: Mid-Chapter quiz p.788

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34


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Lesson 7: Angle Measures and Segment Lengths (PH text 12.4)Objective: to find the measures of angles formed by chords, secants and tangents.

to find the lengths of segments associated with circles

Secant – a line (or segment, or ray) that intersects a circle at two points – a secant always contains a chord

Theorem 12-13 The measure of an angle formed by two lines (or chords) that intersect inside a circle is half the sum of the measures of the intercepted arcs.

Theorem 12-14The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs.

Examples: Find the value of the variables.

1. 2.

3. 4.

36

55

x

35

x90170

35y130

103

39

y


5. 6.

Theorem 12-15 Summary:Case I: The products of the chord segments are equalCase II: The products of the secants and their outer segments are equalCase III: The product of a secant and its outer segment equals the square of the tangent

Examples:7. 8.

9. 10. What is the length of ?

37

z

72

80C

D

B

x55A

7

59

x 11

12

x

16

z

15

10

BA

12

3


11.Find the value of the variables.

Practice:

38

6

8

11y

x

3

2


HW: p.794 #8-20, 24-26Extra Practice:

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Lesson 8: Circles in the Coordinate Plane (PH text 12.5)Objective: to write the equation of a circle

to find the center and radius of a circle

Think about some points that would make this equation true.

Imagine a compass connecting these points in a circular shape.

Theorem 12-16 The standard form of an equation of a circle with center (h, k) and radius r is

This is easily derived from the distance formula.

Examples: Find the center and radius.

1. x2 + y2 = 9 2. x2 + y2 = 36

3. x2 + y2 = 121 4. (x – 6)2 + (y – 2)2 = 25

5. (x – 8)2 + (y + 4)2 = 225 6. (x + 3)2 + (y + 7)2 = 100

Example 7: Write the standard equation of each circle. a) center (3, -4) and radius 6 b) center (-2, -1) and radius

Using Algebra:43


When given the center and a point on the circle, …The distance formula can be used to find the circle’s radius.The midpoint of the diameter is the center of the circle.

What is the standard equation of the circle with center (1, -3) that passes through the point (2, 2).distance formula to find radius:

Use radius (computed) and center (given) to write an equation.

Example 8: A diameter of a circle has endpoints (-3, 7) and (5, 5). Write an equation of the circle.

Practice:

HW: p.801 #8-38 even

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Circle Relationships

Visualize at a glance the relationships between the various angles, arcs, and sectors of a circle.

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Documents

Name:____________________________ - North … · Web viewLesson 7: Angle Measures and Segment Lengths (PH text 12.4) Lesson 8: Circles in the Coordinate Plane (PH text 12.5) Lesson