Circles Notes.notebook

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February 27, 2020

Mar 27­7:04 PM

Parts of a circle (10­6) equation of a circle (12­5) Angles with circles: (12­3) Central angles inscribed angles arcs (12­2) Segments with circles: (12­1, 12­4) tangents secants chords Arc length (10­6) Area of a sector (10­7) Area of a segment of the circle (10­7) Locus (12­6)

D.1.e ­ Locate, describe, and draw a locus in a plane or space D.3.a ­ Identify and define line segments associated with circles (e.g., radii, diameters, chords, secants, tangents) D.3.b ­ Determine the measure of central and inscribed angles and their intercepted arcs D.3.c ­ Find segments lengths, angle measures, and intercepted arc measures formed by chords, secants, and tangents intersecting inside and outside circles D.3.d ­ Solve problems using inscribed and circumscribed polygons F.1.d ­ Find arc lengths and circumferences of circles from given information (e.g., radius, diameters, coordinates) F.1.e ­ Find the area of a circle and the area of a sector of a circle from given information (e.g., radius, diameters, coordinates) G.1.d ­ Write equations for circles in standard form and solve problems using equations and graphs

HSG­CO.A. Experiment with transformations in the planeHSG­CO.A.1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.HSG­C.A. Understand and apply theorems about circlesHSG­C.A.1. Prove that all circles are similar.HSG­C.A.2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. HSG­C.A.4. (+) Construct a tangent line from a point outside a given circle to the circle.HSG­C.B. Find arc lengths and areas of sectors of circlesHSG­C.B.5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.HSG­GPE.A. Translate between the geometric description and the equation for a conic sectionHSG­GPE.A.1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Are all circles similar?What are ways to use the parts of a circle to solve problems?How can you prove relationships between angles and arcs in a circle?When lines intersect a circle or within a circle, how do you find the measure of resulting angles, arcs and segments?How does the length of part of a circle's circumference relate to the central angle of the circle?How does the area of part of a circle formed by the radius and an arc relate to the central angle of a circle?How do you find the equation of a circle in the coordinate plane?How can a locus be used to sketch a geometric relationship?

ChordSecantTangentPoint of tangencyCenterDiameterRadiusConcentric circlesStandard form of a circle

Circle ­ in a plane, the set of all points (a locus) that is equal distance from a fixed point Arc ­a part of the circle Chord ­ a segment whose endpoints are on the circleInscribed angle ­ an angle whose vertex of the angle is on the circle and the sides are chords of the circleSecant ­ a line, ray or segment that intersects a circle at two pointsTangent ­ a line, segment or ray in the plane of the circle that intersects the circle in exactly one point (point of tangency)

Apr 7­8:48 AM

Warm­Up ­ Graffitti WallCircles ­ write everything that you know about circles on the board.

ReviewIf cos(A) = .52, and sin(B)=.52, what can we say about angles A & B?

Distance Formula

Midpoint formula

Mar 28­11:48 AM

with center (h, k) and radius r. (h,k)

r

7.1 ­ Equations of Circles

G.1.d ­ Write equations for circles in standard form and solve problems using equations and graphs

d = √(x2-x1)2 + (y2-y1)2

Mar 28­12:20 PM

Find the center of the circle and its radius. Keep in simplified radical form.

Write the Equations of the following Circles:

Example 1:Center: (3, ­2), radius: 4

Example 2:Center: (4, 5) and radius: √7

1)

Example 3:Center: (1,­3) and point on circle (2,2)

Why will knowing a point on the circle help out??

HW: pg. 800 #7, 14­22(even), 26­32 (even), 38, 44, 45, 52

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February 27, 2020

Feb 28­12:12 PM

Equations of Circles MATH LIB

Jan 26­8:26 AM

Write the equation of the circle in standard form.

Jan 26­8:26 AM

Write the equation of the circle in standard form.

Mar 28­11:57 AM

HW: pg. 800 #7, 14­22(even), 26­32 (even), 38, 44, 45, 52

Warm­Up1. What is the equation of a circle?

2. Can you find the equation of a circle given the diameter? How?

3. What is the formula for Circumference and Area of a circle?

4. A farmer's plot of land was struck by a meteorite which damaged a circular area of his farm. If the farmer's house is labeled as the origin of the coordinate plane, the area damaged by the meteorite can be expressed by the equation:

a. Graph the problem

b. given the following items locations, which were damaged by the meteorite?

Silo (2,4) ; barn (4,6) ; pigpen (8,9).

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February 27, 2020

Feb 12­1:06 PM Feb 13­9:25 AM

Mar 28­2:26 PM

7.2: Parts of a Circle

Circle ­ in a plane, the set of all points (a locus) that is equal distance from a fixed point called the center . (Circles are named by its center.)Diameter ­ segment that contains the center of the circle and has both endpoints on the circle.Radius ­ segment that has one endpoint at the center and the other endpoint on the circle.Congruent Circles ­ have congruent radii.Central Angle ­ angle whose vertex is at the center

ReviewShania is standing at the top of a castle turret in Ireland. Her parents are on the ground below, waving at her 60ft away from the base of the turret. If the turret is 52 feet tall, what is the angle of depression?

Write the standard equation of the circle below.

Mar 28­2:40 PM

An arc is a part of a circle.

Semi­circle ­ half of a circleMinor Arc ­ smaller than a semicircle (named w/ 2 letters)Major Arc ­ larger than a semicircle (named with 3 letters)

1. Name 4 minor arcs of Circle O.

2. Name 2 semicircles.

3. Name 2 major arcs that contain point A.

Adjacent Arcs ­ arcs of the same circle that have exactly one point in common.

Concentric Circles - coplanar circles that have the same center.

What else can we say about these circles?

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February 27, 2020

Mar 30­5:34 PM

7.2: Parts of a Circle (Continued)Arc measures are the same as the central angles it comes from. Arcs added up should equal 360 degrees.

Solve for a:

Apr 2­9:10 AM

7.3 Angles in CirclesArc ­a part of the circle Chord ­ a segment whose endpoints are on the circleInscribed angle ­ an angle whose vertex of the angle is on the circle and the sides are chords of the circleSecant ­ a line or ray that intersects a circle at two pointsTangent ­ a line, segment or ray in the plane of the circle that intersects the circle in exactly one point (point of tangency)

Inscribed Angle Theorem:

Intercepted arc­ an arc of a circle having endpoints on the sides of an angle

Mar 30­9:29 PM

Type of AngleLocation of

VertexHow to find its

measure

Central Angle at the center = intercepted arc

Inscribed Angle

on the circle = ½ (intercepted arc)

7.3 Angles in Circles

Find the measure of a and b.

Mar 30­8:54 PM

Closing Activity

pg. 654 #16­22(even), 28, 40­42(even)

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February 27, 2020

Jan 22­9:08 PM

Warm­Up ReviewIn right triangle ABC, m<C=90 degrees, cos A = 4/5. What is sin B?

The bicycle wheel shown at the right travels 63in in one complete rotation. If the wheel rotates only 120o about the center, how far does it travel? Justify your reasoning!

Feb 14­1:20 PM

Mar 30­8:57 PM

Example 1:

m<RST =

m SU =

S

R

U

T40o

120o

Inscribed angle = 1/2 (intercepted arc)

Let's find all inscribed angles and arcsand find properties of inscribed quadrilaterals.

Mar 30­9:33 PM

(2x + 3)o

(75 ­ 2x)o

D

G

E F

Find m<EDF.

Theorem 12­12: The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.

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February 27, 2020

Apr 1­9:35 AM

Thm: The measure of an angle formed by two lines that intersect inside a circle is half the sum of the measures of the intercepted arcs.

Thm: The measure of an angle formed by two lines that intersect outside the circle, is half the difference of the measures of intercepted arcs.

Jan 29­10:20 PM

A

B

D

E

750

960

1280

Find the missing angle and arc measures.

1.

Jan 29­10:13 PM

Warm­UpRewrite the equation of the circle in standard form.

ReviewBlack Widow is trying to determine the height of a building. She is looking up at an angle of 62 degrees and she is 40 feet from the base of the building. If Black Widow is 5 foot 7 inches, find the height of the building.

xoyo

78o

30o

170o

Find x and y

Feb 20­9:38 AM

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February 27, 2020

Feb 20­9:47 AM Feb 20­9:26 AM

Feb 19­1:06 PM Mar 30­9:43 PM

7.4 Chords and Arcschord ­ a segment whose endpoints are on a circle

Within a circle or in Congruent Circles:(all of these and their converses are true)

1. congruent central angles have congruent arcs.

2. Congruent central angles have congruent chords.

3. Congruent Chords have congruent arcs.

4. Chords equidistant from the center(s) are congruent.

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February 27, 2020

Mar 30­9:59 PM

In the diagram Circle O ≅ Circle P. Given that BC≅DF, what can you conclude?

Ex. 2: Find the measure of arc AD

Example 3: Find x

Mar 30­10:03 PM

Thm: In a circle, if a diameter is perpendicular to a chord, then it bisects that chord and its arc.

Thm: In a circle, if a diameter bisects a chord, (that is not a diameter), then it is perpendicular to the chord.

Thm: In a circle, the perpendicular bisector of a chord contains the center of the circle.

Feb 25­6:04 PM

Activity

Mar 30­10:03 PM

Find the length of the chord.

6.8

x4

Closing Activity

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February 27, 2020

Jan 29­10:28 PM

Warm­Up4.

5.

Find the length of the chord.

6.8

x4

Feb 21­1:33 PM

Feb 21­1:38 PM Mar 14­2:50 PM

Scavenger Hunt Continued

Quiz time!

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February 27, 2020

Mar 15­11:46 AM

Warm­Up

Mar 31­3:55 PM

7.5 Tangents in CirclesTangent to a circle - a line in the plane of the circle that intersects the circle in exactly one point.Point of Tangency - the point where a circle and a tangent intersect. A

B

Thm: A tangent is perpendicular to the radius or diameter of a circle.

X

Y

Z

A

Ex. AB and BC are tangent to F. Find the value of x.

115o

x

A

B

C

F

Ex 2: What is the radius of the circle?

You Try:

Theorem: If a line is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle.EX:

Mar 31­4:32 PM

Thm: If two tangents originate from the same exterior point, then they are congruent.

AB ≅ AC A

B

C

Example 5: Find the value of x.

Apr 1­6:55 AM

Closing Activity

pg. 767 #8­16 (even) (12.1)

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February 27, 2020

Feb 1­2:00 PM

Warm­Up

Apr 1­9:53 AM

Type of Lines Where meet? Picture Rule

2 chords inside

2 secants outside

1 secant, 1 tangent outside

part(part) = part(part)

outside(whole) =outside(whole)

outside(whole) =tan2

7.6 Secants and Chords

Apr 1­10:07 AM

Using the diagram on the Left

1. If BE= 4, CE=3, and DE=9, then AE= __________. E

A

B

C

D

1. If AC= 14, BC = 6, and EC= 21, then CD= ______.

2. IF CD= 5, DE= 7, BC= 4, then AC = _________.

E

A

B

C

D

12

X

166

4

X

4x + 35

5x

6U

X

Y

V

W 21

x + 3

x ­ 4

G

H

I

J

K

Apr 1­10:23 AM

Closing Activity

pg. 794 #8­24 (even), 32

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February 27, 2020

Feb 5­9:30 PM

Warm­Up1. Find the circumference and area of the circle. Leave in terms of pi.

Mar 17­8:33 PM

You and three of your friends have decided to order a pizza. Luckily, the local pizza restaurant is having a deal. You can buy two medium 10­inch pizzas for $7.99 each, or one large 16­inch pizza for $15.99. The plan is to split the pizza four ways. Which order gives you the most pizza?

Mar 30­9:10 PM

arc length: pg. 654 #30­34 (even), 44

7.7 Arc Length and Area of a Sector

Apr 1­10:32 AM

Area of Sector of a circleSector - a region bounded by an arc of the circle and the two radii to the arc's endpoints.

Ex. Find the area of sector ACB. Leave your answer in terms of pi.

A

C

B100o6 m

G

H

L

m GH = 80; LH = 6m

area of sector GLH

In circle L,

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February 27, 2020

Apr 1­10:42 AM

Segment of a circle - a piece of the circle that is bounded by an arc and the segment joining its endpoints.

Area of a Segment

P J

KArea of the shaded green if:PJ = 8 cm

Example:

Ex. Find the area of the shaded region. Round your answer to the nearest tenth.

A

B O120

o 24 ft

Feb 5­9:43 PM

Constructing TangentsConstructing the tangent at a point on a circle.

O

P1. Draw a straight line from the center O, through the given point P and on beyond P.

2. Put the compasses' point on P and set it to any width less than the distance OP. Then, on the line just drawn, draw an arc on each side of P.

3. Now, construct a line perpendicular to point P.

0°

86

Feb 5­9:43 PM

Constructing TangentsConstructing the tangent at a point on a circle.

O

P1. Draw a straight line from the center O, through the given point P and on beyond P.

2. Put the compasses' point on P and set it to any width less than the distance OP. Then, on the line just drawn, draw an arc on each side of P.

3. Now, construct a line perpendicular to point P.

0°86

Jan 22­8:38 PM

Constructing a tangent to a point not on a circle.

P

O

1. Construct a straight line from point O to point P.

2. Construct the perpendicular bisector of line OP (to find the midpoint).

3. From the midpoint of OP, set the compass width to the circle center. Make an arc across the circle at two places.

4. Where the arcs on the circle are made, connect lines and make them to point P.

0°156

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February 27, 2020

Jan 22­8:38 PM

Constructing a tangent to a point not on a circle.

P

O

1. Construct a straight line from point O to point P.

2. Construct the perpendicular bisector of line OP (to find the midpoint).

3. From the midpoint of OP, set the compass width to the circle center. Make an arc across the circle at two places.

4. Where the arcs on the circle are made, connect lines and make them to point P.

0°152

Mar 17­9:10 PM

1. The face of the clock in our classroom has a diameter of 12 inches. Calculate the arc length covered by the minute hand in a 90 minute class period.

2. Grandma has decided to make a blueberry pie. If the standard pie dish has a diameter of 9 inches and the pie is cut into six equal pieces, what is the area of one piece of pie?

3. Which formula should be used to solve the following problem:

What is the distance a Ferris Wheel travels if the radius is 150 ft and it only rotates 60 degrees?

Mar 17­9:14 PM

Work on your performance task!

Apr 1­2:18 PM

1 What is the area of sector CD

A 6.105 cm2

B 10.144 cm2

C 25.399 cm2

D 21.369 cm2

pg. 663­664 #16­26 (even), 42(10.7)

Closing

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February 27, 2020

Feb 7­10:02 AM

Warm­UpSolve for x

Mar 21­1:01 PM

Mar 22­1:51 PM Mar 22­1:56 PM

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February 27, 2020

Mar 22­9:13 AM Feb 5­10:30 PM

Practice these!

Jan 22­8:38 PM

Solve for x

Feb 13­8:36 AM

x

26

Parking LotSolve for x

Have you review sheet out on your desk!

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February 27, 2020

Mar 27­1:23 PM Feb 5­10:30 PM

Warm­Up1. 2.

Feb 5­10:34 PM

Practice SAT/ACT problems

1. 2.

3.4.

5. 6.

7. 8.

1. B 5. E

2. C 6. C

3. D 7. D

4. D 8. C

Feb 5­10:53 PM

9.

9. K 13. K

10. D 14. B

11. G 15. B

12. K 16. H

10.

11.

12.

13.

14.

15.

16.

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February 27, 2020

Feb 5­10:30 PM

Closing

Any last questions before the test tomorrow

Feb 16­8:20 AM

Apr 23­9:24 PM

Locus(or loci)is the set of all points that satisfy one or more conditions.

Try This

What is the locus of points that are 5 cm from a given point?

1. Choose a point on your paper and label it P.

2. Using a ruler, measure 5 cm away from pt P (in any direction) and plot a point.

3. Continue to choose different directions from pt P and plot 12 more points that are 5 cm from pt P.

4. What figure do the points seem to form?

The most common locus of points is a CIRCLE.

The set of points that are a distance r from a point is a locus of points called a circle centered at the given point with a radius of r.

r Locus of points

Apr 23­9:26 PM

Try This

What is the locus of points that are 5 cm from line l ?

1. Draw a horizontal line l.

2. Using a ruler, measure 5 cm starting at the line measuring straight up and plot a point.

3. Continue to measure from the line at different locations on the line plotting 10 more points.

4. Do the same thing below the line l.

5. What is actually formed?

The next common locus is the set of points at a fixed distance d from a line (l).

The set of points that are a distance d from a given line is a set of lines parallel to the given line from a given distance d.

Locus of points

d

d

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February 27, 2020

Apr 23­9:27 PM

Try This

What is the locus of points that is the same distance from point A and point B?

1. Draw point A and point B.

2. Using a straight edge, draw the segment AB.

3. Plot a point (label the point G) on the segment that is equidistant from point A and point B.

[Point G is the ___________ of the segment AB]

4. Plot points that are directly above and below point G. Use a ruler and check that the points are equidistant from point A and point B.

5. What do the points form?

The third common locus of points is the set of points that are equidistant from two given points.

The set of points that are equidistant from two given points, A and B is a line which is the perpendicular bisector of the segment with endpoints of A and B.

Locus of points

A B

Apr 23­9:28 PM

The locus of points equidistant from two intersecting lines is the angle bisectors of each pair of vertical angles formed by the lines.

The locus of points that are equidistant from two parallel lines, is a line midway between the two parallel lines.

(*The dotted lines represent the locus of points)

Apr 23­9:29 PM

A locus of points satisfying two conditions

EXAM

PLE

Point P is in the interior of <ABC. What is the locus of points in the interior of <ABC that are equidistant from both sides of <ABC and 2 inches from P?

Solution

The locus of points equidistant from both sides of <ABC is the angle bisector.The locus of points 2 inches from P is a circle.

The locus of points that satisfy both conditions is the intersection of the angle bisector and the circle.

The locus can be 2 points, 1 point or 0 points. (Depends on the location of P)

Apr 23­9:29 PM

REAL­WORLD The epicenter of an earthquake is the point on Earth's surface that is

directly above the earthquake's origin.

A seismograph can measure the distance to the epicenter, but not the direction to the epicenter.

To locate the epicenter, readings from three seismographs in different locations are needed.

The reading from seismograph A tells that the epicenter is somewhere on the circle centered at A.

The reading from B tells that the epicenter is one of two points of intersection of circle A and circle B.

The reading from C tells which of the two points of intersection is the epicenter (the locus of points)

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February 27, 2020

Apr 23­9:31 PM

TRY IT!!

#1 Which term describes the locus of points in a plane that are equidistant from (4, 3) and (2, 1)?

A) Circle

B) Line

C) Plane

D) Point

Zero

the

Her

o

ZB

#3 Which point lies on the locus of points equidistant from (2, 3) and (­5, 3)?

A) (6, ­1.5)

B) (­1.5, 6)

C) (1.5, 1)

D) (1, 1.5)

#2 Given line AB perpendicular to plane M, what is the locus of points 4 inches from both line AB and plane M?

A) 2 points

B) 2 lines

C) 2 circles

D) 4 points

M

A

B

Each circle has a radius of 4 inches, with a center contained on line AB

andeach circle is 4 inches away from plane M.

ClickandReveal­Square

Click to reveal explanation of answer

http://www.regentsprep.org/Regents/math/geometry/MultipleChoiceReviewG/Locus.html

Apr 1­2:25 PM

11.8 Locus: A Set of Points

Locus­ a set of points, all of which meet a stated condition.(Loci is the plural of locus)

We can also use locus descriptions for geometric terms:

For example, the locus of points in the interior of an angles that are equidistant from the sides of the angle is an angle bisector.

Or ­ in a plane, the locus of points that are equidistant from a segments endpoints is the perpendicular bisector of the segment.

CW/HW: pg. 808­810 #7­9, 31­34, 37­41

Apr 23­9:12 PM

Warm­Up

x

26

Apr 5­8:53 PM