Chapter 8: Circle Geometry

May 07, 2012

Chapter 8: Circle Geometry

8.1 Properties of Tangents to a Circle

What do you already know about circles?

May 07, 2012


You will need a compass, a ruler, and protractor for this unit.

Problem page 384

What angle does the spoke appear to make with the ground?

Point of Tangency

tangent line


Tangent: A line that intersects the circle at only one point.

Point of Tangency: The point of intersection between a circle and a tangent line.

A

B Which line is a tangent? A or B?

A line may look as if it is a tangent to a circle but it may not be. How can you determine if the line isa tangent?

May 07, 2012

A tangent to a circle is always perpendicular to the radius at the point of tangency

A

B

P

O

APO = BPO = 90o


We will use pythagorean theorem to solve problems:

a2 + b2 = c2

3 m

4m


May 07, 2012

25o


Also remember that all angles of a triangle will

add up to 180o

What is the missing angle?

0

63º

A

B

Ex. 1 Point O is the center of the circle. and AB is a tangent to the circle. In ▲OAB, ∠AOB = 63 º. Determine the measure of ∠OBA.


May 07, 2012

Example 2 pg. 386

C

D

O

15 cm

20 cm

Determine the length of the radius to the nearest tenth.

Which line is the tangent? What angle does it make with the radius? Which side,then,is the hypotenuse?


An airplane,A, is cruising at an altitude of 9000m. A cross section of Earth is a circle with radius approximately 6400 km. A passenger wonders how far she is from a point H on the horizon she sees outside the window. Calculate the distance to the nearest kilometer.

A

H

O

Earth

6400 km

9000m


May 07, 2012

OnlineTangentApplet & Pratice

Homework:

# 3, 4, 5, 6, 7, 8, 13, 16c,

for enrichment try #11, 19 and 20


8.2 Properties of Chords in a Circle

A

BActivity page 392

1. Trace a circle using half of a petri dish

2. Choose two points on your circle and draw a straight line between them. Make sure your line DOES NOT go through the centre of the circle

3. Label your line segment AB. This line is called a chord

4. Cut out your circle

5. Fold your circle so that A coincides with B. Open your circle and draw a straight line along the crease that was formed.

6. Label the point where your crease line intersects AB as point C

A

B

C

What observations can you make? About the angles formed? About the length of AC and CB?

What if you repeat it with two different points?

May 07, 2012

Chord Property 1 - The perpendicular from the center of a circle to a chord bisects the chord.

Chord Property 2 - The perpendicular bisector of a chord in a circle passes through the center of a circle.

Chord Property 3 - A line that joins the center of a circle and the midpoint of a chord is perpendicular to the chord.


A cord is a line segment that joins two points on a circle.

Ex. 1 Find the unknown angles:

A

BC

O

xºyº

33º

* recognize the radius and its relationship to the isosceles triangle!

Point O is the center of the circle. OC bisects chord AB.


May 07, 2012

O

A

B

C D E

26 cmdiameter =

10 cm

Example 2: Find the length of CD


O

A

B

C DE

26 cmdiameter =

10 cm

Example 3:Find the length of CD.


May 07, 2012

A BC

D

O

20 cm

Example 4: Problem Solving

A horizontal pipe has a circular cross section,with center O. Its radius is 20 cm. Water fills less than one half of the pipe. The surface of the water AB is 24 cm wide. Determine the maximum depth of the water which is depth CD.


Assignment ~ pages 397 - 399# 3, 4, 5, 6, 7, 10, 11, 17, 18

* Note: # 10, 17, and 18 may need to be reviewed as an opener for next class


May 07, 2012

8.3 Properties of Angles in a Circle

Have students discover the central angle theory through Investigate on page 404 (materials required - compass, ruler, protractor)

1. Choose two points on a circle and label them A and B

2. Choose a third point C and connect AC and BC

3. Label the centre of your circle O and connect AO and BO

4. Measure angle AOB and Angle ACB

What do you notice? Compare with your classmates

O

A

B

C

Definitions:

Major Arc - The longer of the 2 arcs between 2 points on a circle.

Minor Arc - The shorter of the 2 arcs between 2 points on a circle.

A B


May 07, 2012

Inscribed Angle - an angle formed by joining the end points of an arc to a POINT ON THE CIRCLE.

Central Angle - An angle formed by joining the end points of an arc to the CENTER OF THE CIRCLE.


P

O

Q

R<POR =2<PQR

Circle Angle Property #1:In a circle, the measure of a central angle subtended by an arc is twice the measure of an inscribed angle subtended by the same arc.


May 07, 2012


A

B

C

D

E

1. Choose two points on a circle and label them A and B

2. Choose a third point C and connect AC and BC

3. Choose a fourth point D and connect AD and BD

4. Measure angles ACB and ADB

What do you notice? Compare with your classmates. Try a third inscribed angle

Circle Angle Property #2In a circle, all inscribed angles subtended by the same arc are congruent.


A

B

C

D

E

<ACB = <ADB = <AEB

May 07, 2012

A

BF

GH


Circle Angle Property #3All inscribed angles subtended by a semi circle are right angles

Example 1

Putting the theories to work ....... like solving puzzles

A B

C

D

O

xº

yº55º

(pg. 407)

* Use Inscribed and Central Angles


May 07, 2012

A

B

C

D8.5 cm

5.0 cm

Example 2 - Use Angle Inscribed in a SemicircleFind the length of the rectangle to the nearest tenth.


Example 3 ~

Determine Angles in an Inscribed Triangle A

xº

zº

120º

120ºO

B

C

yº


May 07, 2012

Assignment ~ page 410 - 412

# 3 - 6, 8,11,12

8.4 - Cyclic Quadrilaterals

A. Have students complete the investigation on page 415. Discusstheir discoveries.

Materials required: compass, ruler, protractor

May 07, 2012

Discovery ~

Opposite angles of an inscribed quadrilateral have a sum of 180º

When two angles have a sum of 180º,they are called SUPPLEMENTRY ANGLES.

May 07, 2012

a

2a

a

2ab2b

May 07, 2012

O140º

xº

yº

O

11.1 cm

12.0 cm

Find the perimeter of the hexagon.

May 07, 2012

O

11.1 cm

12.0 cm

find the area of the hexagon

Assign questions from pgs. 419 - 421

Chapter 8: Circle Geometry

Documents

Transcript of Chapter 8: Circle Geometry