11 .1 Lines that intersect Circles - Sarah Smithmrsklap.weebly.com/uploads/3/0/4/4/30442020/geometrych11...Chapter 11.4 Inscribed Angle Definitions - complete the definitions below

Geometry: Chapter 11 Note Packet Name: __________________________

Page 1 of 14

11 .1 Lines that intersect Circles

Definitions & Examples

Interior of a circle- all pts circle. Ex.

Exterior of a circle- all pts. Circle. Ex.

Chord- a whose endpoints are on the circle.

Ex.

Secant- a line that intersects a circle in

Ex.

Tangent of a circle- a line that intersects a circle in

Ex.

Point of tangency- the point where the touches the .

Ex.

Congruent circles- two or more circles with radii.

Concentric circles- circles with the same .

Tangent circles- circles that in point.

Common tangent- one line that is to circles.

Example #1) Use circle A to draw and label the following:

a. a line ℓ that is tangent to circle A at point C

b. a radius AB

c. a chord CB

d. a secant BD

e. a diameter ED

f. a point R that is on the interior of the circle

g. a point S that is on the exterior of the circle

h. a circle Z that is tangent to Circle A at point E

i. a circle W that is tangent to line ℓ

j. a concentric circle X that lies inside Circle W


Page 2 of 14

Theorem Hypothesis (If) Conclusion (Then)

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

Symbols:

A

B

If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle.

Symbols:

A

B

If two segments are tangent to a circle from the same external point, then the segments are congruent.

Symbols:

Ex. 2) AB and BC are both tangent to circle O. Find the value of x and then find the length of BC

Ex. 3) Use the picture at the right to find:

a. the radius of both circles: Circle S = Circle R. =

b. the point of tangency =

c. the equation for the line of tangency

*** Note: The general equation for horizontal lines are y =

The general equation for vertical lines are x =

Ex. 4) The surface of the earth is approximately 4000 miles from the center. If a satellite orbits the earth at 120

miles above the surface find the distance from the satellite to the horizon of the earth.


Page 3 of 14

Chapter 11 .2 Arcs and Chords

Definitions and Examples

Central angle- an angle whose is at the of a circle.

Arc- an part of the circle with two endpoints.

Example: Use Circle X at the r i g h t :

a) Identify a central angle =

b) Identify two arcs =

Complete the definitions of each type of arc and describe their measures:

Def: Minor Arc-

Def: Major Arc –

Major arc = minus the measure of the central angle xo

Def: Semicircle-

Always equals

Def: Adjacent Arcs – The measure of an arc formed by two adjacent arcs = the sum of the measure of the two arcs.

Def: Congruent Arcs -

Two arcs that have


Page 4 of 14

Use the pie chart at the right to find:

a. measure of arc FG

b. measure of arc KLF

c. the measure of GJK

Ex. 2)Use circle F and it’s central angles at the right to find the measure of:

a. arc AE =

b. arc ABD =

c. arc AE + arc ED =

d. arc ABC =

e. arc CD =

f. the measure of central AFB

Ex. 3 Use circle D at the right to give examples of the following (use proper arc notation)

a. A minor arc:

b. A major arc:

c. A semicircle: B

d. A radius:


Page 5 of 14

Def: Congruent arcs- are or more arcs with the same .

Ex 1) Use the above theorem to answer the following questions:

If HLG JLK, then what two statements can you make regarding chords HG and JK and arcs HG

and JK? Write the true statements below:

What must the value of y equal?

Ex 2) Given TV SW

find the value of n.

Ex 3) Circle B and E are congruent and so are arcs AC and DF. Find m DEF.


Page 6 of 14

12y + 20

10y + 36


Page 7 of 14

Theorem Hypothesis Conclusion

If a radius is perpendicular to

a chord, then it the chord

and its

The perpendicular bisector of

a chord is the

Ex 4) If RS = 8 and SM = 9 find the length of chord NP

Hint: the radius of circle R is the hypotenuse of a right triangle that can be used to find NS.

Ex. 5) Find the length of chord CE


Page 8 of 14

Chapter 11 .3 Sector Area and Arc Length

Recall how to find the Area of a Circle: A = r 2

Definitions and Examples

Area of a Sector is a fraction of the containing the

sector.

Sector of a circle- is a region bound by radii and their

intercepted _.

Write the formula to find the Area of a Sector with a central angle mo

Read example 1 on page 764. The answer the following: Ex.

1) Find the area of the sectors to the nearest tenth.

a. b.

Segment of a circle- is a region bound by an and its _.

Segment of a circle

Area of a segment of a Circle = Area of sector – Area of triangle

Recall how to find the area of a triangle: A =

b

h

2

Area of a Sector =


Page 9 of 14

Example 2) Find the area of the segment.

Recall how to find the circumference of a circle: C = d or C= 2 r

The Arc length is a fraction of the of a circle.

Arc length- is the measured along an

in linear units.

Ex 3) Find the length in centimeters of the arc FG.

Ex. 4) Find the length of the arc BD to the nearest tenth.

Segment


Page 10 of 14

Chapter 11.4 Inscribed Angle

Definitions - complete the definitions below and use the picture to give an example of each definition.

Inscribed angle- an whose

vertex is the circle and whose sides are

of the circle.

Intercepted arc- is the arc whose points lie on the

sides of the angle and all the

points between them.

Subtends- a chord or arc whose endpoints are on the

sides of an angle.

Study and complete the Theorems and their conclusions.

= inscribed angle

= intercepted arc

= subtends

Inscribed Angle Theorem: The

measure of an inscribed angle is

the measure of its

intercepted arc.

A

C

B

Conclusion:

Corrollary: If inscribed angles of a

circle intercept the arc

or are subtended by the same chord

or arc, then the angles are

.

A

D

C

F

B

Conclusion:

Inscribed Semicircle: An inscribed angle subtends a semicircle if and

only if the angle is .

C

A B

conclusion

Inscribed Quadrilateral: If a

quadrilateral is inscribed in a circle

then its

angles are

.

Conclusion:


Page 11 of 14

Use the Inscribed Angle Thm. to answer the following. Ex1)

Find the measure of arc PS and inscribed angle PRU

Use the Corollary of the Inscribed Angle Thm. to answer the following

Ex. 2) Find the m LJM and arc LM

Use the Inscribed Semicircle Thm. to answer the following

Ex3) Find the value of a.

(5a +20)

Use the Inscribed Quadrilateral Thm. to answer the following Example 4:

Find the measures of the angles in the quadrilateral.


Page 12 of 14

1020

1

900

Then the measure of the angle is :


Chapter 11.5 Angle Relationships in Circles

**Study and complete the theorems of this section**

IF THE VERTEX OF THE ANGLE IS…

Theorem 11-5-1

ON the circle

This could be two chords, two secants, or

one tangent and one secant

Theorem 11-5-2

INSIDE the circle

This could be two secants or two chords.

Theorem 11 – 5- 3

OUTSIDE the circle

This could be two secants, one secant and one tangent or

two tangents.


1120

mDCB

mC


Page 13 of 14

Ex 1) Find the measure of FGH

Ex 2) Find the measure of arc LM

Ex 3: Find m ABD.

Ex 4) Find the measure of arc KM

Ex 5) find the measure of ACD


Page 14 of 14

Chapter 11.7 Circles in the Coordinate Plane

The equation of a circle formula is: where r stands

for , h stands for and k stands for .

Ex 1: State the center and radius of the circle.

a. (x-1)2 + (y+2)2 = 9 b. (x+ 4)2 + (y-2)2 = 34

Ex 2: Graph the circle in Ex. 1 pt. a.

Ex 3: List four other points on the circle in part a.

How could you determine those points without relying on the graph?

Ex 4: What are the equations of the circles graphed below?

–2

–3

–4

–5

–6

–7

–1

–1

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11 .1 Lines that intersect Circles - Sarah Smithmrsklap.weebly.com/uploads/3/0/4/4/30442020/geometrych11...Chapter 11.4 Inscribed Angle Definitions - complete the definitions below