GEO: Unit 8 – Circles NAME _____________________________

12.1 – Lines that Intersect Circles (1)

Objectives: Identify tangents, secants, and chords. Use properties of tangents to solve problems.

This photograph was taken 216 miles above Earth. From this altitude, it

is easy to see the curvature of the horizon. Facts about circles can help us

understand details about Earth.

Recall that a circle is the set of all points in a plane that are ____________________ from a

given point, called the ______________of the circle.

A circle with center C is called circle C, or C.

The of a circle is the set of all points inside the circle. The ______________

of a circle is the set of all points outside the circle.

Example 1: Identify each line or segment that intersects L.

chords: secant: diameter:

tangent: point of tangency: radii:

Example 2: Identify each line or segment that intersects P.

chords: secant: diameter:

tangent: point of tangency: radii:

Example 3: Find the length of each radius. Identify the point of tangency and write the

equation of the tangent line at this point.

Radius of S =

Radius of R =

Point of tangency =

Equation of tangent line:

A __________ ____________ is a line that is tangent to two circles.

Example 4: Early in its flight, the Apollo 11 spacecraft orbited Earth at an altitude of 120 miles. What

was the distance from the spacecraft to Earth’s horizon rounded to the nearest mile? Draw and label a

diagram to support your answer. (It is 4000 miles from the center of the earth to the surface….radius.)

Example 5: Kilimanjaro, the tallest mountain in Africa, is 19,340 ft tall. What is the distance from the

summit of Kilimanjaro to the horizon to the nearest mile? Draw and label a diagram to support your

answer. (Miles and feet…convert to miles. 5280 ft = 1 mile)

Example 6: RS and RT are tangent to Q. Find RS.

12.1 HOMEWORK: page 797 #1-4, 6-11, 13-27, 31.

12.2 Notes – Arcs and Chords (2)

Objectives: Apply properties of arcs. Apply properties of chords.

A __________________________is an angle whose vertex is the center of a circle.

An _____________is an unbroken part of a circle consisting of two points called

the endpoints and all the points on the circle between them.

Minor arcs may be named by two points. Major arcs and semicircles must be named by three points.

Example 1: The circle graph shows the types of

grass planted in the yards of one

neighborhood.

Find mKLF .

Example 2: Use the graph to find each of the following.

a. m FMC

b. mAHB

c. m EMD

__________________________ are arcs of the same circle that intersect at exactly one point.

RS and ST are adjacent arcs.

Example 3: Find mBD

Within a circle or congruent circles,

_______________ __________ are two arcs that have the

same measure. In the figure ST UV .

Example 4: TV WS . Find mWS .

Example 5: C J, and m GCD m NJM . Find NM.

Example 6: Find NP. (RS = 8 and SM = 9.)

Example 7: Find ZY. Round to the nearest hundredth.

12.2 HOMEWORK: page 806 #1-28, 31, 38, 39, 51

12.3 – Sector Area and Arc Length (3)

Objectives: Find the area of sectors. Find arc length.

Warm up:

1. What is the area of a rectangle 12 in long and 6 in wide?

2. What is the area of half of the rectangle in number 1?

3. What is the area of ¾ of the rectangle in number 1?

4. What is the area of a circle with radius of 8 miles?

5. What is the area of half of a circle with radius of 6 kilometers?

Example 1: Find the area of each sector. Give answers in terms of π and rounded to the nearest hundredth.

sector HGJ

Example 2: Find the area of each sector. Give answers in terms of π and rounded to the nearest hundredth.

sector ABC

Example 3: Find the area of each sector. Give your answer in terms of π and rounded to the nearest hundredth.

sector ACB

Example 4: A windshield wiper blade is 18 inches long. To the nearest square inch, what is the area covered by

the blade as it rotates through an angle of 122°?

In the same way that the area of a sector is a fraction of the area of the circle, the length of

an arc is a fraction of the circumference of the circle.

Example 5: Find each arc length. Give answers in terms of π and rounded to the nearest

hundredth.

Find FG .

Example 6: Find each arc length. Give answers in terms of π and rounded to the nearest hundredth. An arc with

measure 62o in a circle with radius 2 m

Example 7: Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth.

Find GH .

12.3 HOMEWORK: page 813 #1-5, 9-11, 13, 15, 23, 26, 30, 32.

12.4 – Inscribed Angles (5)

Objectives: Find the measure of an inscribed angle. Use inscribed angles and their properties to solve

problems.

An _____________ ____is an angle whose vertex is on a circle and whose sides contain chords of the

circle.

An ________________ consists of endpoints that lie on the sides of an inscribed angle and all the points

of the circle between them.

A chord or arc _______________ an angle if its endpoints lie on the sides of the angle.

Example 1: Find each measure.

m PRU

mSP

Example 2: Find each measure.

m DAE

mADC

Example 3: Find a.

Example 4: Find mLJM.

Example 5: Find z.

Example 6: Find mEDF.

Example 7: Find the angle measures of quadrilateral GHJK.

Example 8: Find the angle measures of quadrilateral JKLM.

12.4 HOMEWORK: page 824 #1-11, 16, 17-19, 21, 26.

12.5 - Angle Relationships in Circles (6)

Objectives: Find the measures of angles formed by lines that intersect circles.

Use angle measures to solve problems.

Example 1: Using Tangent-Secant and Tangent-Chord Angles

Find each measure.

m EFH

mGF

Example 2: Using Tangent-Secant and Tangent-Chord Angles

Find each measure.

m STU

mSR

I

Finding Angle Measures Inside a Circle

Example 3: Find each measure.

m AEB

m AED

Example 4: Find each angle measure.

m ABD

m ABE

Example 5: Find each angle measure.

m RNM

Finding Measures Using Tangents and Secants

Example 6: Find the value of x.

Example 7: Find the value of x.

Example 8: In the company logo shown, 108mFH , and 12mLJ

What is m FKH ?

Example 9: Two of the six muscles that control eye movement are attached to the eyeball and intersect behind

the eye. If mAEB = 225, what is mACB?

Angle Relationships in Circles

Where is the vertex of the angle?

What is the measure of the angle?

Draw a diagram to depict the angle and circle.

Conclusion:

Finding Arc Measures

Example 10: Find mYZ

Example 11: Find mLP

12.5 HOMEWORK: page 834 #1-17, 21, 25, 27-30.

12.7: Circles in the Coordinate Plane (7)

Objectives: Write equations and graph circles in the coordinate plane.

Use the equation and graph of a circle to solve problems.

REVIEW: What is the distance formula?

Use that formula to find the distance between (2, 2) and (5, 7).

Example 1: Write the equation of each circle below:

a) J with center J(2, 2) and radius 5.

b) P with center P(0, –3) and radius 8

c) K that passes through J(6, 4) and has center K(1, –8)

d) Q that passes through (2, 3) and has center Q(2, –1)

What is the connection between the distance formula above and the equation of the circle?

If you are given the equation of a circle, you can graph the circle by making a table or by identifying its center

and radius.

Graph x2 + y2 = 16.

Step 1

Step 2

OR YOU CAN…

Example 2: For each of the following circle equations, list the center and the radius.

1. 2 2 2( ) ( )x h y k r 2.

2 2 36x y

x

y

C =

r =

C =

r =

x

y

3. 2 2( 3) ( 3) 4x y 4.

2 2( 3) ( 3) 1x y

5. 2 2( 2) 81x y

6. 2 2( 7) 9x y

Graph (x – 3)2 + (y + 4)2 = 9. Graph x² + y² = 9.

Graph (x – 3)2

+ (y + 2)2

= 4

12.7 HOMEWORK: page 850 #1-8, 10-17, 19, 20, 22-26, 30, 33-35.

x

y

x

y

x

y

C =

r =

C =

r =

C =

r =

C =

r =