lesson 3.1 properties of tangents - mrs....

CCGPS UNIT 3 – Semester 1 ANALYTIC GEOMETRY of 32

Circles and Volumes

Name: _________________

Date: ________________

Understand and apply theorems about circles MCC9-12.G.C.1 Prove that all circles are similar. MCC9-12.G.C.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. MCC9-12.G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove

properties of angles for a quadrilateral inscribed in a circle. MCC9-12.G.C.4 (+) Construct a tangent line from a point outside a given circle to the circle. Find arc

lengths and areas of sectors of circlesMCC9-12.G.C.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 e angle as the constant of proportionality; derive the formula for the area of a sector. Explain volume formulas and use them to solve

problems MCC9-12.G.GMD.1 Give formal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use

dissection arguments, Cavalieri’s principle, and informal limit arguments. MCC9-12.G.GMD.2 (+) Give an informal argument using Cavalieri’s principle for the

formulas for the volume of a sphere and other solid figures. MCC9-12.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Lesson 3.1 Properties of Tangents

Goal: Use properties of a tangent to a circle.

A circle is the set of all points in a plane that are equidistant from the center of a circle.

A radius is a segment whose endpoints are the center and any point on the circle.

A chord is a segment whose endpoints are on the circle.

A diameter is a chord that contains the center of a circle. The diameter is twice the radius.

A secant is a line that intersects the circle in two points.

A tangent is a line in the plane of the circle that intersects the circle in one point only.

Theorem 1: A tangent line will be perpendicular to a radius of the circle at the point of tangency.

Theorem 2: Tangent segments from common external points are congruent.

PROBLEMS Based on the definitions above draw the following lines.

1. Radius 2. Chord 3. Diameter

4. Secant 5. Tangent Define: AB___________

PC___________

XY___________

B

A P Y

X C

Note: P is the center of the circle


PROBLEMS: Tell how many common tangents the circles have by drawing them.

1.

2.

3.

Example 1: Draw common tangents. Tell how many common tangents the circles have by drawing them.

a. b.

Solution:

a. b.

You can fit 3 common tangents. You can fit 1 common tangent.


A

PROBLEMS: Verify a tangent to a circle. Show that CB is tangent to the circle (or nota tangent to the circle).

4.

B AB=10, AC=6, CB=8

C

5.

B AB= 340 , AC=12, CB=14

C

6.

B AB=14, AC=7, CB=9

C

Example 2: Verify a tangent to a circle. Show that ED is tangent to the circle.

D AD=17, AE=8, BD=15

E

Solution:

To prove that ED is tangent to the circle with radius AE you must show that AE is perpendicular

to ED (Theorem 1). Therefore, apply Pythagorean theorem:

2 2 2

2 2 217 8 15

289 64 225

289 289

AD AE ED

Since the left side equals the right side, we have a right triangle and ED is a tangent.

A

A

A


A r

R

PROBLEMS: B is the point of tangency. Find the radius of the circle.

7.

A

r r

B 36

48

8.

B 28

r

A r 14

Example 3: B is the point of tangency. Find the radius of the circle.

r

B 50

70 C

Solution: From Theorem 1 you know that AB is perpendicular to the tangent BC. Use the Pythagorean

theorem to solve for the radius, r:

2 2 2

2 2 2

2 2

2 2

( 50) 70

( 50)( 50) 70

100 2500 4900

100 2400

24

AC AB BC

r r

r r r

r r r

r

r


B

A A

B

PROBLEMS: AC is tangent to the circle at A and BC is tangent to the circle at B. Find the value of x.

9.

25

C

3 10x

10.

A

2x

C

16

Example 4: RS is tangent to the circle at S and RT is tangent to the circle at T. Find the value of x.

S

24 R

T 2x+4

Solution: RS RT Tangent segments from the same point are congruent.

24 2 4x Substitute.

10x Solve for x.


Lesson 3.2 Properties of Circles

Circles and their Relationships among Central Angles, Arcs, and Chords In this unit you will study properties of circles. As you progress through the unit, you will learn new definitions and theorems. In some cases you will discover, or develop and prove these theorems. It may be advantageous for you to keep a “Circle Book” or list that includes the definitions and theorems addressed in each task. With each definition and theorem you enter, you should also include an illustrative sketch. We will begin by re-visiting the definition of a circle.

Circles are identified by the notation ○P, where P represents the point that is the center of the circle.

A central angle of a circle is an angle whose vertex is at the center of the circle. ∠APB is a central angle of ○P.

A portion of a circle’s circumference is called an arc. An arc is defined by two endpoints and the points on the circle

between those two endpoints. If a circle is divided into two unequal arcs, the shorter arc is called the minor arc and the

longer arc is called the major arc. If a circle is divided into two equal arcs, each arc is called a semicircle. A diameter cuts

a circle into two equal arcs. In our figure below, we call the portion of the circle between and including points A and B,

arc AB notated by AB . We call the remaining portion of the circle arc ACB , or .ABC Note that major arcs are usually

named using three letters. The central angle APB has the same measure (in degrees) as the arc AB . Note that when

we refer to the arc of a central angle, we usually mean the minor arc unless otherwise stated.

PROBLEMS: Fill in the missing terms.

A ____________________________ of a circle is an angle whose vertex is the center of the circle

A _______________________ is an arc with endpoints that are the endpoints of a diameter.

Using the picture below: An _______________ is portion of circumference also known as an unbroken part of a circle. If

180 ,ACB then the points on the circle that lie on the interior of ACB form a _______________________ with

endpoints A and .B This minor arc is named _______. The points on the circle that do not lie on the minor arc form a

______________________ with endpoints A and .B The major arc is named ___________.

A

D C B

Now we will introduce some notation and terminology needed to study circles. Consider the figure at right. m∠APB = 75˚


PROBLEMS: Find the measure of each arc. EB is the diameter.

A

B 125˚ 25˚ C D

E

1. AB 2. AC 3. CE

4. EB 5. CEB 6. CBE

7. CA 8. ABE 9. EBA

Example 1: Find the measure of each arc of the circle with center D , diameter AB and 40 .BDC

a. .AB b. .AC c. .CBA

Solution: A D B

a. AB is a semicircle and therefore 180 . C

b. 40 .BDC That means that 40BC as well. Therefore, 180 40 140 .AC

c. CBA is BC and .AB 40 180 220 .CBA


Find the measure of each arc. EB is a diameter.

A

B 95˚ 20˚ C D

E 70˚

F

10. CB 11. FB 12. CE

13. BFC 14. CF 15. EBA

Find the measure of each arc. EB is a diameter.

A

B 107˚ 13˚ C D 100˚

E

F

16. CE 17. EF 18. EBA

19. ABC 20. CEB 21. AF


Lesson 3.3 Properties of Cords

PROBLEMS: A C

1. If 15, .AB find CD

2. If 105 , .AB find CD

B D

3. If 20 60 , .AC and BD find AB

4. If 20 95 , .AC and AB find BD

5. If 25 90 , .AC and CD find BD

Goal: Use relationships between arcs and chords in a circle.

Theorem 1: In the same circle two minor arcs are congruent if and only if their A C

congruent chords are congruent. AB CD if and only if AB CD .

Example: If 95 , .AB CD then AB CD

B D

Theorem 2: If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. A

If a diameter of a circle is perpendicular to a chord, then the diameter bisects the B chord and its arc. C E

If CBis to AD , then .AE EDand AB BD D

Example: If 6, 6 .If CBis to ADand AE then ED and AB BD

Theorem 3: In the same circle two chords are congruent if and only if they are C equidistant from the center. A M

.AB CDif and onlyif KL KM L K D

Example: 8, 8.If KL KM and AB thenCD B


Problems:

In the diagram in example 2, suppose 16 6.AB and MO

4. Find CD

5. Find AM

6. Find ON

In the diagram in example 2, suppose 24, 2 4 10.AB MO x and ON x

7. Find CD

8. Find x

9. Find ON

Example 2: 12. 3 7 8. .AB CD MO x and ON x Find MO C

A N

M O D

Solution: B

Chords AB and CD are congruent, therefore they are equidistant from the center, .O That means .MO ON

MO ON By theorem 3.

3 7 8x x Substitute.

2x Solve for x.

So 3 3(2) 6.MO x


Find the measure of the given arc or chord.

10. BC A

116

C B

11. AB A

A B

C B

12. AC

A

D B

13

C

13. Find x.

12x+7

A

D B

C

3x+16


Lesson 3.4 Inscribed Angles and Polygons

Goal: Use inscribed angle of circles.

A inscribed angle is an angle whose vertex is on a circle and whose sides B contain chords of a circle. Inscribed Angle

A C

Theorem 1: The measure of an inscribed angle is one half the measure of its intercepted arc.

Example: If 40BAC then 2 40 80 .BC

Theorem 2: If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

Example: If 100AB then 50ADB ACB A

D

B C

Theorem 3: If a right angle is inscribed in a circle, then the hypotenuse is a diameter of a circle. C

Example: Since the right angle ACB is inscribed in the circle, the B

hypotenuse is the diameter, AB.

A

Theorem 4: A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

Example: 180 .A D 180C B 95 A B 105

95 2 180y 105 180x

C x 2y D


PROBLEMS: Use Theorems 1 and 2 to solve the problems.

A

9. What is AB ? 35

B

10. What is AB ? A

48 D

11. What is DE ?

B 41 E

12. If 70 ,NO what is ?NPO M

N

13. If 106 ,MP what is ?MOP

P O

14. Which two pairs of angles are congruent?

15. Find the measure of the indicated angle or arc:

a. B b. D

A 63˚ P C A P C

115˚

E

Find .AB Find .CAE


Use Theorem 4 to solve the problems.

16. Find x and y.

115˚

120˚

4x˚

5y˚

17. Find x and y.

2x˚ 8y˚

y˚+80˚ x˚

18. Find x and y.

5x˚ 2y˚

8y˚ x˚


Lesson 3.5 Other Angle Relationships in Circles

Goal: Find the measures of angles inside or outside a circle. C

Theorem 1:If a tangent and a chord intersect at a point on the circle, then 220˚ 1 B the measure of each angle formed is one half the measure of its intercepted arc.

Example: If 220BCA then 220

1 110 .2

A

Theorem 2: If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

Example: If 86AB and 78CD then 86 78

82 .2

x

A D

86˚ x˚ x˚ 78˚

B C

Theorem 3: If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one-half the difference of the measures of the intercepted arcs.

A C

Example: If 125CD and 25AB , then 115 25

452

x

x˚ 25˚ 115˚

B D

PROBLEMS: Use Theorem 1 to solve the problems.

170 x A

1. What is the measure of ?x

B

2. What is the measure of ?x

x 70 A

B


Use Theorem 2 to solve the problems. A 75 B

3. What is the measure of ?x x

C D

85

x B

4. What is the measure of ?x A 120

C D

100

B

5. What is the measure of ?x A

98 74 x

C D

B

6. What is the measure of ?x A x

98 74

C D


Use Theorem 3 to solve the problems. B

7. What is the measure of ?x A 125

x 25

C D

A

8. What is the measure of ?x 185

x 45

C D

9. What is the measure of ?x 285

x


Lesson 3.6 Segment Lengths in Circles

Goal: Find segment lengths in circles. C

Theorem 1: If two chords intersect in the interior of a circle, then the products B of the lengths of the segments of one chord is equal to the products of the O lengths of the other chord. A

In other words: AO BO CO DO D

Theorem 2: If two secant segments share the same endpoint outside a circle, then the product of the lengths of one entire secant segment and its external segment equals the product of the lengths of the other entire secant segment and its external segment.

In other words: EA EB EC ED B

A

E C

D

Theorem 3: If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the entire secant segment and its external segment equals the square of the length of the tangent segment. A

In other words: 2EA EC ED E C D

Example 1:

Find OB. C

Solution: Use Theorem 1. 8 B 6 O x A 10

AO BO CO DO

6 8 10x

13.3x


PROBLEMS:

Use Theorem 1 to solve the problems. A B

1. What is the measure of x? 8 12

x 6

C D

2. What is the measure of x? A B

7 10

5 x

C D

3. What is the measure of x? B

20

A 16 x D

16

C


Use Theorem 2 to solve the problems.

4. What is the measure of x? x

6

7 10

5. What is the measure of ?x x 22

4

8

Example 2:

Find EB

Solution: Use Theorem 2.

EA EB EC ED B

( 9) 4 (4 5)x x x A 9

2 9 36x x E 4 C 5 D

2 9 36 0

( 12)( 3) 0

12 3

x x

x x

x and x

Bu substitution: Length cannot be negative, so throw out x = -12.

EB = x + 9 =3 + 9 = 12


6. What is the measure of x? x

5 10

7. What is the measure of x?

4 9

x

Example 3:

Find EC.

Solution: Use Theorem 3.

A

2EA EC ED 6

26 ( 9)x x E x C 9 D

2

2

36 9

9 36 0

( 12)( 3) 0

12 3

x x

x x

x x

x and x

Length cannot be negative, so EC = x = 3.


Lesson 3.7 Circumference and Arc Length

PROBLEMS: Find the indicated measure.

1. The circumference of a circle with a radius of 12 meters.

2. The radius of a circle with a circumference of 25 feet.

3. The diameter of a circle with a circumference of 50 meters.

4. The radius of a circle with a diameter of 10 feet.

Goal: Find arc length and other measures of circles.

The circumference of a circle is the distance around the circle.

An arc length is a portion of the circumference of a circle.

Theorem 1: The circumference C of a circle is 2C r or ,C d where r = radius and d = diameter.

In a circle the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360˚.

Example 1:

a. Find the circumference of a circle with a radius of 5 meter.

Solution: 2C r

2 5 31.4 .C meter

b. Find the diameter of a circle with a circumference of 20 meters.

Solution: C d

20 d

20

6.4d meters


PROBLEMS: Find the length of .AB

5. A 6. A 7.

90˚ 105˚ 197˚

15 ft B 7m B A 12 cm B

Example 2:

Find the length of .AB B

Radius = 5 meters. 130˚ A

Solution: 130

2 5 360

AB

The ratio of the arc to the circumference is the same as

the ratio of the angle to 360˚

130

2 5 11.4 .360

AB meters

Solve for .AB



8. mAB 9. .Radiusof E 10. .Circumferenceof E

A T

9 in B 45 cm 170˚

15 ft E 55˚ 2 m E

B A S

Example 3:

Find the circumference of .E B

C 260˚ E

81.6 cm A

Solution: 260

360

ACB

Circumference

The ratio of the arc ACB (81.6 cm) to the circumference is the

same as the ratio of the angle to 360˚

360

81.6 113 .260

Circumference cm

Solve for the circumference.


Find the perimeter of the region.

11. 20cm

20 cm r=10 cm

12.

Radius = 3 cm 20 cm

7 cm

13. radius = 5 cm

15 cm


Lesson 3.8 Areas of Circle and Sectors


1. The diameter of a circle is 12 centimeters. Find the area.

2. The area of a circle is 22 square meters. Find the radius.

3. The area of a circle is 500π square meters. Find the diameter.

Goal: Find the areas of circles and sectors.

A sector of a circle is the region bounded by two radii of the circle and their intercepted arc.

Theorem 1: The area of a circle is times the square of the radius. ( 2A r )

Theorem 2: The ratio of an area of a sector of a circle to the area of the whole circle is equal to the ratio of the measure of the intercepted arc to 360˚.

Example 1: Find the indicated measure.

a. Radius b. Area

r 5 in

Area= 20 m²

Solution: 2A r 2A r

220 r 25A

20

2.53r

78.5A


PROBLEMS: Find the area of the lined sector.

4.

21 cm

250˚

5.

16 in

175˚

6.

13 cm

75˚

Example 2: Find the area of the lined sector.

11 in

165˚

Solution: 165

360

Shaded Area

Entire Areaof Circle

By Theorem 2

2

165

360 11

Shaded Area

Substitute

2 216511 174

360Shaded Area in

Solve


Lesson 3.9 Surface Areas and Volumes of Spheres

PROBLEMS:

1. Find surface area and volume of a sphere with a radius of 12 cm.

2. Find surface area and volume of a sphere with a diameter of 15 cm.

3. Find radius of a sphere with a surface area of 315 m².

4. Find the diameter of a sphere with a volume of 1220 cubic inches.

Goal: Find the surface areas and volumes of spheres.

A sphere is the set of all points in space which are equidistant from a given point (= center of sphere). A typical example of a sphere is a round ball.

Theorem 1: The surface area, S, of a sphere is: 24S r

Theorem 2: The volume, V, of a sphere is: 34

3V r

Example 1: Find surface area and volume of a sphere with a radius of 8 inches.

Solution:

Surface Area Volume

2

2

2

4

4 8

804

S r

in

3

3

3

4

3

48

3

2144.7

V r

in


Lesson 3.10 Cavalieri and Volumes of Cylinders, Pyramids, and Cones

Problems:

Now use 10 congruent rectangles to approximate the figure (remember l = 1cm)

1. What is the length of each rectangle?

2. What is the height of each rectangle?

3. What is the area of each rectangle?

4. What is the area of 10 rectangles?

Cavalieri’s principle: If, in two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal (Kern and Bland 1948, p. 26).

Both coin stacks have the same volume

since the sections made by planes

parallel to and at the same distance

from their respective bases are always

equal.

You could also say they have the same

amount of identical coins (2 cent Euro

coins).

That is pretty much Cavalieri’s principle

in simple terms.

At any given height, the horizontal length of the

figure on the left is 1 cm. In problems 1 – 5 you

will calculate the figure’s area.

http://mathworld.wolfram.com/Volume.html


5. Suppose you use 100 rectangles

a. What is the area of each rectangle in that case?

b. What is the area of all rectangles in that case?

6. As n gets larger, do the rectangles approximate the figure better, worse, or the same?

Problems:

7. Find volume of a cylinder radius of 5 cm and a height of 3 cm.

8. What is the height of a cylinder with a volume of 50 cm³ and radius of 3 cm?

9. What is the radius of a cylinder with a volume of 3 m³ and a height of 2 m?

Example 1: Find volume of a cylinder radius of 2 inches and a height of 6 inches.

Solution: The formula for the volume of a cylinder is 2.cylinderV base height wherebase r

Plug the numbers into formula and solve for the unknown variable:

2 2 32 6 75.4cylinderV base height r height in


Problems:

10. Find volume of a pyramid with a base length of 5 cm and a height of 20 cm.

11. What is the height of a pyramid with a volume of 50 cm³ and base length of 3 cm?

12. What is the base area of a pyramid with a volume of 3 m³ and a height of 1 m?

Example 2: Find volume of a pyramid with a square base of 2 inches and a height of 6 inches.

Solution: The formula for the volume of a pyramid is 21.

3pyramidV b h


2 2 32 6 75.4cylinderV base height r height in


Problems:

13. What is the volume of an ice cream cone (which you can buy from Ms. Bonnell) with a radius of 1 inch and a

height of 3 inches?

14. What is the radius of a cone with a volume of 3 m³ and a height of 2 m?

15. What is the height of a cone with a volume of 100 m³ and a radius of 2 m?

Example 3: Find volume of a cone with a radius of 2 inches and a height of 6 inches.

Solution: The formula for the volume of a cone is 21.

3coneV r h


2 2 31 12 6 25.1

3 3coneV r h in

lesson 3.1 properties of tangents - mrs....

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