Geometry: All-In-One Answers Version A

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Geometry: All-In-One Answers Version A

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Lesson 1-1

Lesson Objective 1 Use inductive reasoning to make

conjectures

Patterns and Inductive Reasoning

NAEP 2005 Strand: Geometry Topic: Mathematical Reasoning Local Standards: ____________________________________

Vocabulary. Inductive reasoning is reasoning based on patterns you observe.

A conjecture is a conclusion you reach using inductive reasoning.

A counterexample is an example for which the conjecture is incorrect.

Examples.

1 Finding and Using a Pattern Find a pattern for the sequence. 384

192

96

48

Use the pattern to find the next two terms in the sequence.

384, 192, 96, 48, . . . Each term is half

the preceding term. The next two

2 2 2

terms are 48 2 24 and 24 2 12 .

2 Using Inductive Reasoning Make a conjecture about the sum of the cubes of the first 25 counting numbers.

Find the first few sums. Notice that each sum is a perfect square and that the perfect squares form a pattern.

13

1 12 12

13 23

9 32 (1 2)2

13 23 33

36 62 (1 2 3)2

13 23 33 43

100 102 (1 2 3 4)2

13 23 33 43 53 225 152 (1 2 3 4 5)2

The sum of the first two cubes equals the square of the sum of the first two counting numbers. The sum of the first three cubes equals the

square of the sum of the first three counting numbers. This pattern continues for the fourth and fifth rows. So a conjecture might be that

the sum of the cubes of the first 25 counting numbers equals the square of the sum of the first 25 counting numbers, or (1 2 3 . . . 25)2.

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3 Finding a Counterexample Find a counterexample for the conjecture. Since 32 42 52, the sum of the squares of two consecutive numbers is the square of the next consecutive number.

Begin with 4. The next consecutive number is 4 1 5 .

The sum of the squares of these two consecutive numbers is 42 52 16 25 41 .

The conjecture is that 41 is the square of the next consecutive number. The square of the next consecutive number is (5 1)2 62 36 . So, since 42 52 and 62 are not the same, this conjecture is false .

4 Applying Conjectures to Business The price of overnight shipping was $8.00 in 2000, $9.50 in 2001, and $11.00 in 2002. Make a conjecture about the price in 2003.

Write the data in a table. Find a pattern.

2000 $8.00

2001 $9.50

2002 $11.00

Each year the price increased by $ 1.50 the price in 2003 will increase by $ 1.50 be $11.00 $ 1.50 $ 12.50 .

. A possible conjecture is that . If so, the price in 2003 would

Quick Check. 1. Find the next two terms in each sequence.

a. 1, 2, 4, 7, 11, 16, 22, 29 , 37 , . . .

b. Monday, Tuesday, Wednesday, Thursday , Friday , . . .

c. Answers may vary. Sample:

,

,...

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2. Make a conjecture about the sum of the first 35 odd numbers. Use your calculator to verify your conjecture.

1

1 12

1 3

4 22

1 3 5

9 32

1 3 5 7

16 42

1 3 5 7 9 25 52

The sum of the first 35 odd numbers is 352, or 1225.

3. Finding a Counterexample Find a counterexample for the conjecture. Some products of 5 and other numbers are shown in the table.

5 7 35 5 3 15 5 11 55

5 13 65 5 9 45 5 25 125

Therefore, the product of 5 and any positive integer ends in 5.

Since 5 2 10, not all products of 5 end in 5. (But they all either end in 5 or 0.)

4. Suppose the price of two-day shipping was $6.00 in 2000, $7.00 in 2001, and $8.00 in 2002. Make a conjecture about the price in 2003.

Each year the price increased by $1.00. A possible conjecture is that the price in 2003 will increase by $1.00. If so, the price in 2003 would be $8.00 $1.00 $9.00.

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Lesson 1-2

Lesson Objectives 1 Make isometric and orthographic

drawings 2 Draw nets for three-dimensional

figures

Drawings, Nets, and Other Models

NAEP 2005 Strand: Geometry Topic: Dimension and Shape Local Standards: ____________________________________

Vocabulary.

An isometric drawing of a three-dimensional object shows a corner view of the figure drawn on isometric dot paper. An orthographic drawing is the top view, front view, and right-side view of a three-dimensional figure. A foundation drawing shows the base of a structure and the height of each part.

A net is a two-dimensional pattern you can fold to form a three-dimensional figure.

Examples. 1 Isometric Drawing Make an isometric drawing of

the cube structure at right.

An isometric drawing shows three sides of a figure from a corner view. Hidden edges are not shown, so all edges are solid lines.

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2 Orthographic Drawing Make an orthographic drawing of the isometric drawing at right.

Orthographic drawings flatten the depth of a figure. An orthographic drawing shows three views. Because no edge of the isometric drawing is hidden in the top, front, and right views, all lines are solid.

Front Right

Front

Top

Right

3 Foundation Drawing Make a foundation drawing for the isometric drawing.

To make a foundation drawing, use the top view of the orthographic drawing.

Front Right

Top

Because the top view is formed from 3 squares, show 3 squares in the foundation drawing. Identify the square that represents the tallest part. Write the number 2 in the back square to indicate that the back section is 2 cubes high. Write the number 1 in each of the two front squares to indicate that each front section is 1 cube high.

2

Right

1

1

Front

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4 Identifying Solids from Nets Is the pattern a net for a cube?

If so, name two letters that will be opposite faces.

A

The pattern

is

a net because you

can

fold it to form a cube. Fold squares A and C up to form the back

and front of the cube. Fold D up to form a side. Fold E over to

form the top. Fold F down to form another side.

BDE F C

After the net is folded, the following pairs of letters are on opposite faces: A and C are the back and front faces.

B and E are the bottom and

top

faces.

D and F are the right and left side faces.

5 Drawing a Net Draw a net for the figure with a square base and four isosceles triangle faces. Label the net with its dimensions.

Think of the sides of the square base as hinges, and "unfold" the figure at these edges to form a net. The base of each of the four isosceles triangle faces is a side of the square . Write in the known dimensions.

10 cm

8 cm

8 cm 10 cm

Quick Check. 1. Make an isometric drawing of the cube structure below.

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Name_____________________________________ Class____________________________ Date ________________ 2. Make an orthographic drawing from this isometric drawing.

Front

Top

Right

Front Right

3. a. How many cubes would you use to make the structure in Example 3? 4

b. Critical Thinking Which drawing did you use to answer part (a), the foundation drawing or the isometric drawing? Explain. Answers may vary. Sample: the foundation drawing; you can just add the three numbers.

4. Sketch the three-dimensional figure that corresponds to the net.

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5. The drawing shows one possible net for the Graham Crackers box.

14 cm

7 cm

20 cm CGRRAACHKAEMRS

7 cm 14 cm

CGRRAACHKAEMRS

20 cm

Draw a different net for this box. Show the dimensions in your diagram.

Answers may vary. Example:

14 cm

20 cm

7 cm

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Lesson 1-3

Lesson Objectives 1 Understand basic terms of geometry 2 Understand basic postulates of

geometry

Points, Lines, and Planes

NAEP 2005 Strand: Geometry Topic: Dimension and Shape Local Standards: ____________________________________

Vocabulary and Key Concepts.

Postulate 1-1

Through any two points there is exactly one line.

t

B

Line t is the only line that passes through points A and B .

A

Postulate 1-2

If two lines intersect, then they intersect in exactly one point.

AC B

D

E

*) *)

AE and BD intersect at C .

Postulate 1-3 If two planes intersect, then they intersect in exactly one line.

R ST W

*)

Plane RST and plane STW intersect in SSTT .

Postulate 1-4 Through any three noncollinear points there is exactly one plane.

A point is a location. Space is the set of all points. A line is a series of points that extends in two opposite directions without end. Collinear points are points that lie on the same line.

t B A

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A plane is a flat surface that has no thickness. Two points or lines are coplanar if they lie on the same plane.

B AC Plane ABC

A postulate or axiom is an accepted statement of fact.

Examples.

1 Identifying Collinear Points In the figure at right,

X

name three points that are collinear and three points

that are not collinear.

Points Y , Z , and W lie on a line, so they are

collinear.

mZ

Any other set of three points in the figure do not lie on a line,

so no other set of three points is collinear. For example, X, Y,

and Z form a

triangle

and are not collinear.

YW

2 Naming a Plane Name the plane shown in two different ways.

R

U

You can name a plane using any three or more points on that plane

that are not collinear.

Some possible names for the plane shown are the following:

S

T

plane RST , plane RSU , plane RTU ,

plane STU , and plane RSTU .

3 Finding the Intersection of Two Planes Use the diagram at right. What is the intersection of plane HGC and plane AED?

As you look at the cube, the front face is on plane AEFB, the back face

is on plane HGC, and the left face is on plane AED. The back and left

faces of the cube intersect at HD . Planes HGC and AED intersect

vertically at

*)

HD

.

H E

D A

G F

C B

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4 Using Postulate 1-4 Shade the plane that contains X, Y, and Z.

Z

Points X, Y, and Z are the vertices of one of the four triangular

faces of the pyramid. To shade the plane, shade the interior of the triangle formed by X , Y , and Z .

Y

X

Quick Check. 1. Use the figure in Example 1.

a. Are points W, Y, and X collinear? no

V W

b. Name line m in three different ways.

* )* )* )

Answers may vary. Sample: ZW , WY , YZ .

c. Critical Thinking Why do you t*hink) arrowheads are used when drawing a line or naming a line such as ZW ?

Arrowheads are used to show that the line extends in opposite directions without end.

2. List three different names for plane Z. Answers may vary. Sample: HEF, HEFG, FGH.

*)

3. Name two planes that intersect in BF . ABF and CBF

H

G

Z

E

F

H E

G F

D A

C B

4. a. Shade plane VWX. Z

Y V

W

b. Name a point that is coplanar with points V, W, and X. Y

X

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Lesson 1-4

Lesson Objectives 1 Identify segments and rays 2 Recognize parallel lines

Segments, Rays, Parallel Lines and Planes

NAEP 2005 Strand: Geometry Topic: Relationships Among Geometric Figures Local Standards: ____________________________________

Vocabulary.

A segment is the part of a line consisting of two endpoints and all points between them.

Segment AB

AB

A

B

Endpoint

Endpoint

A ray is the part of a line consisting of one endpoint and all the points of the line on one side of the endpoint.

)

Ray YX

YX X

Y

Endpoint

Opposite rays are two collinear rays with the same endpoint.

Parallel lines are coplanar lines that do not intersect.

Q

R

S

)

RQ

and

)

RS

are opposite rays.

Skew lines are noncoplanar; therefore, they are not parallel and do not intersect.

D A

H

E

C B

G

F

AB is

parallel

to EF.

AB and CG are skew lines.

Parallel planes are planes that do not intersect.

A D

G J

H

B

Plane ABCD is

parallel

to plane GHIJ.

I

C

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Examples.

1 Naming Segments and Rays Name the segments and rays in the figure.

A

The labeled points in the figure are A, B, and C.

A segment is a part of a line consisting of two endpoints and all points

between them. A segment is named by its two endpoints. So the

segments are

BA (or AB)

and

BC (or CB)

.

A ray is a part of a line consisting of one endpoint and all the points of B

C

the line on one side of that endpoint. A ray is by any other point on the ray. So the rays are

named)

BA

by

iatns dendpoBiCn)t

first, .

followed

2 Identifying Parallel and Skew Segments Use the figure at right. Name all segments that are parallel to AE. Name all segments that are skew to AE.

Parallel segments lie in the same plane, and the lines that contain them do not intersect. The three segments in the figure that are parallel to AE are BF , CG and DH .

D

A H

E

Skew segments are segments that do not lie in the same plane. The four segments in the figure that do not lie in the same plane as AE are BC ,

CD , FG and GH .

3 Identifying Parallel Planes Identify a pair of parallel planes in your classroom.

Planes are parallel if they do not intersect . If the walls of your classroom are vertical, opposite walls are parts of parallel planes. If the ceiling and floor of the classroom are level, they are parts of parallel planes.

C B

G F

Quick Check.

))

1. Critical Thinking Use the figure in Example 1. CB and BC form a line. Are

they opposite rays? Explain.

No; they do not have the same endpoint.

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2. Use the diagram in Example 2. a. Name all labeled segments that are parallel to GF. HE, CB, DA

b. Name all labeled segments that are skew to GF. AB, DC, AE, DH

c. Name another pair of parallel segments and another pair of skew segments. Answers may vary. Sample: CG, BF; EF, DH

3. Use the diagram to the right. a. Name three pairs of parallel planes.

PSWT RQVU, PRUT SQVW, PSQR TWVU

S

Q

P

R

W

V

T

U

*)

b. Name a line that is parallel to PQ .

*)

TV

c. Name a line that is parallel to plane QRUV.

*)

Answers may vary. Sample: PS

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Lesson 1-5

Lesson Objectives 1 Find the lengths of segments

Measuring Segments

NAEP 2005 Strand: Measurement Topic: Measuring Physical Attributes Local Standards: ____________________________________

Vocabulary and Key Concepts.

Postulate 1-5: Ruler Postulate The points of a line can be put into one-to-one correspondence with the real numbers so that the distance between any two points is the absolute value of the difference of the corresponding numbers.

Postulate 1-6: Segment Addition Postulate If three points A, B, and C are collinear and B is between A and C, then AB BC AC.

A BC

A coordinate is a point's distance and direction from zero on a number line.

the length of AB

A

B

R AB 5 u a b u

a

b

Q coordinate of A

a coordinate of B

Congruent () segments are segments with the same length.

2 cm

A

BA

B

2 cm

AB CD

C

DC

D

AB CD

A midpoint is a point that divides a segment into two congruent segments.

midpoint

A BC

|

|

AB BC

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Examples.

1 Comparing Segment Lengths Find AB and BC.

Are AB and AC congruent?

AB = -5 - (?1) ? = -4? = 4

BC = ?1 - 4? = ??5? = 5

AB BC so AB and AC are

not congruent.

A

B

C

4 3 2 1 0 1 2 3 4 5

.

2 Using the Segment Addition Postulate If AB 25,

2x 6

x 7

find the value of x. Then find AN and NB.

A

N

B

Use the Segment Addition Postulate (Postulate 1-6) to write an equation.

AN NB AB

( ) ( ) 2x 6 x 7 25

Segment Addition Postulate Substitute.

3x 1 25

Simplify the left side.

3x 24 Subtract 1 from each side.

x 8

( ) AN 2x 6 2 8 6 10 ( ) NB x 7 8 7 15

Divide each side by 3 . Substitute 8 for x.

AN 10 and NB 15 , which checks because the sum of the segment lengths equals 25.

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Geometry: All-In-One Answers Version A (continued)

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3 Using the Midpoint M is the midpoint of RT.

5x 9

8x 36

Find RM, MT, and RT.

R

M

T

Use the definition of midpoint to write an equation.

RM MT

Definition of midpoint

5x 9 8x 36 Substitute.

5x 45 8x

Add 36 to each side.

45 3 x

Subtract 5x from each side.

15 x

( ) RM 5x 9 5 15 9 84 ( ) MT 8x 36 8 15 36 84

Divide each side by 3 . Substitute 15 for x.

RT RM MT 168

Segment Addition

Postulate

RM and MT are each 84 , which is half of 168 , the length of RT.

Quick Check.

1. Find AB. Find C, different from A, such that AB and AC are congruent.

A

B

7 6 5 4 3 2 1 0 1 2 3 4 5

AB |7 (2)| |5| 5 We also want BC 5. That means that C must lie 5 units on the other side of B from A. So C must lie at ?2 5 3.

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2. EG 100. Find the value of x. Then find EF and FG.

4x ? 20 2x + 30

E

F

G

x 15, EF 40; FG 60

3. Z is the midpoint of XY, and XY 27. Find XZ. 13.5

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Lesson 1-6

Lesson Objectives 1 Find the measures of angles 2 Identify special angle pairs

Measuring Angles

NAEP 2005 Strand: Measurement Topic: Measuring Physical Attributes Local Standards: ____________________________________

Vocabulary and Key Concepts.

Postula)te 1-7: P)rotractor Postulate

))

Let OA and OB be opposite rays in a plane. O* A ,) OB , and all the rays with

endpoint O that can be drawn on one side of AB can be paired with the

real num) bers from 0 to 180 so that) a. OA is) paired with 0 and OB) is paired with 180

b. If OC is paired with x and OD is paired with y, then

. mlCOD 5 u x 2 y u .

C

60

70 110

80 100

50130 120

90

100 80

110 70

120

60 50130

D

x

y

150

30 150 14040

14040 30

160

10 20

160 20

170

170 10

A0

180 B O

Postulate 1-8: Angle Addition Postulate

If point B is in the interior of AOC, then

m AOB m BOC mAOC.

A

B

OC

If AOC is a straight angle, then mAOB mBOC 180 .

B

AO C

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An angle () is formed by two rays with the same endpoint. The rays are the sides of the angle and the endpoint is the vertex of the angle.

B

T1Q TBQ

x?

acute angle 0 , x , 90

x?

right angle x 5 90

x?

obtuse angle 90 , x , 180

x?

straight angle x 5 180

An acute angle has a measurement between 0 and 90. A right angle has a measurement of exactly 90. An obtuse angle has a measurement between 90 and 180. A straight angle has a measurement of exactly 180. Congruent angles are two angles with the same measure.

Examples.

1 Naming Angles Name the angle at right in four ways.

The name can be the number between the sides of the angle: l3 .

The name can be the vertex of the angle: lG .

Finally, the name can be a point on one side, the vertex, and a point on the other side of the angle: lAGC or lCGA .

3 G

C A

2 Measuring and Classifying Angles Find the measure of each /AOC.

Classify each as acute, right, obtuse, or straight.

a.

b.

C

70 60 110 50130 120

80 100

90

100 80

110 70

120

60 50130

x

y

150

30 150 14040

70

60 50130 120

110

80 100

90

100 80

110 70

120

60 50130

C

150

x

y

14040 30

30 150 14040

14040 30

160

10 20

160 20

160

170

10 20

160 20

170 10

170 10

A0

170

O 60, acute

180 B

A0

O 150, obtuse

180 B

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3 Using the Angle Addition Postulate Suppose that m1 42 and mABC 88. Find m2.

Use the Angle Addition Postulate (Postulate 1-8) to solve.

m 1 m 2 mABC

Angle Addition Postulate

42 m2 88

Substitute 42 for m1 and 88 for mABC.

m2 46

Subtract 42 from each side.

4 Identifying Angle Pairs In the diagram identify pairs of numbered angles that are related as follows: a. complementary 3 and 4

b. supplementary 1 and 2; 2 and 3

c. vertical angles 1 and 3

A

1

2

B

C

12

3 5

4

5 Making Conclusions From a Diagram Can you make each

A

conclusion from the diagram?

a. A C

Yes; the markings show they are congruent.

B b. B and ACD are supplementary

No; there are no markings.

C

D

c. m(BCA) m(DCA) 180 Yes; you can conclude that the angles are supplementary from the diagram.

d. AB BC Yes; the markings show they are congruent.

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Quick Check. 1. a. Name CED two other ways.

l2, lDEC

b. Critical Thinking Would it be correct to name any of the angles E? Explain. No, 3 angles have E for a vertex, so you need more information in the name to distinguish them from one another.

2. Find the measure of the angle. Classify it as acute, right, obtuse, or straight.

60; acute

3. If mDEG 145, find mGEF. 35

G DE F

4. Name an angle or angles in the diagram supplementary to each of the following: a. DOA b. EOB

a. AOB b. BOC or DOE

5. Can you make each conclusion from the information in the diagram? Explain. a. 1 3 b. 4 and 5 are supplementary c. m(1) m(5) 180

a. Yes; the markings show they are congruent. b. Yes; you can conclude the angles are adjacent and

supplementary from the diagram. c. Yes; you can conclude that the angles are

supplementary from the diagram and their sum is 180 by the Angle Addition Postulate.

E

A

D

O

B

C

51 42

3

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Lesson 1-7

Lesson Objectives 1 Use a compass and a straightedge to

construct congruent segments and congruent angles 2 Use a compass and a straightedge to bisect segments and angles

Basic Constructions NAEP 2005 Strand: Geometry Topic: Relationships Among Geometric Figures

Local Standards: ____________________________________

Vocabulary. Construction is using a straightedge and a compass to draw a geometric figure.

A straightedge is a ruler with no markings on it.

A compass is a geometric tool used to draw circles and parts of circles called arcs.

Perpendicular lines are two lines that intersect to form right angles.

A perpendicular bisector of a segment is a line, segment, or ray that is perpendicular to the segment at its midpoint, thereby bisecting the segment into two congruent segments. An angle bisector is a ray that divides an angle into two congruent coplanar angles.

Examples.

1 Constructing Congruent Segments Construct TW congruent

K

to KM.

Step 1 Draw a ray with endpoint T.

T

Step 2 Open the compass the length of KM.

Step 3 With the same compass setting, put the compass point

T

on point T. Draw an arc that intersects the ray. Label the

point of intersection W.

KM TW

A

D

C

B

J

K

N

L

M

W

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2 Constructing Congruent Angles Construct Y so that Y G. Step 1 Draw a ray with endpoint Y.

Y

75

G

Step 2 With the compass point on point G, draw an arc that intersects both sides of G. Label the points of intersection E and F.

E

75

G

F

Step 3 With the same compass setting, put the compass point on point Y. Draw an arc that intersects the ray. Label the point of intersection Z.

Y

Z

Step 4 Open the compass to the length EF. Keeping the same

X

compass setting, put the compass point on Z. Draw an arc

that intersects with the arc you drew in Step 3. Label the

point of intersection X.

)

Step 5 Draw YX to complete Y.

Y

Z

X

Y G Daily Notetaking Guide

Y

Z

Geometry Lesson 1-7 25

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Geometry: All-In-One Answers Version A (continued)

Name_____________________________________ Class____________________________ Date ________________

3 Constructing the Perpendicular Bisector

Given: AB.* )

*)

Construct: XY so that XY AB at the midpoint M of AB.

Step 1 Put the compass point on point A and draw a long arc. Be sure that the opening is greater than 12AB.

A

B

A

B

Step 2 With the same compass setting, put the compass point on point B and draw another long arc. Label the points where the two arcs intersect as X and Y.

*)

*)

Step 3 Draw XY. The point of intersection of AB and XY is M,

the midpoint of AB.

*)

*)

XY AB at the midpoint of AB, so XY is the

perpendicular bisector

of AB.

X

A

B

Y

X

A M

B

Y

*)

4 Finding Angle Measures WR bisects AWB so that mAWR x and mBWR 4x 48. Find mAWB.

W

A

R x

4x 48 B

m AWR m BWR

x 4x 48

3x 48 x 16

mAWR 16

( ) mBWR 4 16 48 16

mAWB m AWR m BWR mAWB 16 16 32

Definition of

angle bisector

Substitute x for mAWR and 4x 48 for mBWR.

Subtract 4x from each side.

Divide each side by 3 .

Substitute 16 for x.

Angle Addition

Postulate

Substitute 16 for mAWR and for mBWR.

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Name_____________________________________ Class____________________________ Date ________________

Quick Check. 1. Use a straightedge to draw XY. Then construct RS so that RS 2XY.

X

Y

R

S

2. a. Construct F with mF 2mB.

B

B

F

)

b. Explain how you can use your protractor to check that YP is the angle bisector of XYZ.

Measure XYP and PYZ to see that they

X

P

are congruent.

Y

Z

3. Draw ST. Construct its perpendicular bisector.

S

T

)

4. If mJKN 50?, then find mNKL and mJKL. KN bisects JKL.

J

K

mNKL 50? , mJKL 100?

NL

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Name_____________________________________ Class____________________________ Date________________

Lesson 1-8

Lesson Objectives 1 Find the distance between two points

in the coordinate plane 2 Find the coordinates of the midpoint

of a segment in the coordinate plane

The Coordinate Plane

NAEP 2005 Strand: Measurement Topic: Measuring Physical Attributes Local Standards: ____________________________________

Key Concepts.

Formula: The Distance Formula The distance d between two points A(x1, y1) and B(x2, y2) is

d

(x2 x1)2 (y2 y1)2

.

Formula: The Midpoint Formula

The coordinates of the midpoint M of AB with endpoints A(x1, y1) and B(x2, y2) are the following:

( x1 x2

M

2

,

) y1 y2 2

.

Examples.

1 Applying the Distance Formula How far is the subway ride from Oak to Symphony? Round to the nearest tenth. Each unit represents 1 mile.

Oak has coordinates (1, 2) . Let (x1, y1) represent Oak. Symphony has coordinates (1, 2) . Let (x2, y2) represent Symphony.

y 6 North

4 Central

Jackson (2, 4)

Cedar2 Symphony

4 2 Oak

(1, 2)

City Plaza x (0, 0)

4 South

d

"(x2 2 x1)2 1 (y2 2 y1)2

Use the Distance Formula. Elm 6

( ( )) ( ( )) d % 1 1 2 2 2 2 Substitute.

d% 2 2 4 2 d % 4 16 %w2w0w

Simplify within parentheses. Simplify.

20

4.472135955

Use a calculator.

To the nearest tenth, the subway ride from Oak to Symphony is 4.5 miles.

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2 Finding the Midpoint AB has endpoints (8, 9) and (6, 3). Find the coordinates of its midpoint M.

Use the Midpoint Formula. Let (x1, y1) be (6, 3) .

(8, 9)

and (x2, y2) be

The midpoint has coordinates

( ) x1 x2 , y1 y2 .

2

2

( ) 8 6

2

The x-coordinate is

2

2 1

Midpoint Formula Substitute 8 for x1 and 6 for x2. Simplify.

The y-coordinate is

( ) 9 3

6

2

2 3

Substitute 9 for y1 and 3 for y2. Simplify.

The coordinates of the midpoint M are (1, 3) .

3 Finding an Endpoint The midpoint of DG is M(1, 5). One endpoint is D(1, 4). Find the coordinates of the other endpoint G.

Use the Midpoint Formula. Let (x1, y1) be (1, 4) and the midpoint

a x1

1 2

x2,

y1

1 2

y2 b

be

(1, 5)

. Solve for

x2

and

y2 , the

coordinates of G.

Find the x-coordinate of G.

1 2

1 x2 2

1 x2

d Use the Midpoint Formula. S d Multiply each side by 2 . S

3 x2

d Simplify. S

4 y2

5

2

10 4 y2

6 x2

The coordinates of G are (3, 6) .

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Geometry: All-In-One Answers Version A (continued)

Name_____________________________________ Class____________________________ Date ________________ Quick Check. 1. a. Using the map in Example 1, find the distance between Elm and Symphony.

about 8.9 miles

b. Maple is located 6 miles west and 2 miles north of City Plaza. Find the distance between Cedar and Maple. about 3.2 miles

2. Find the coordinates of the midpoint of XY with endpoints X(2, 5) and Y(6, 13). (4, 4)

3. The midpoint of XY has coordinates (4, 6). X has coordinates (2, 3). Find the coordinates of Y. (6, 9)

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Lesson 1-9

Lesson Objectives 1 Find perimeters of rectangles and

squares, and circumferences of circles 2 Find areas of rectangles, squares, and

circles

Perimeter, Circumference, and Area

NAEP 2005 Strand: Measurement Topic: Measuring Physical Attributes Local Standards: ____________________________________

Key Concepts.

Perimeter and Area s s

Square with side length s. Perimeter P 4s Area A s2

b

h

h

b Rectangle with base b and height h.

Perimeter P 2b 2h

Area A bh

d

r O

C Circle with radius r and diameter d.

Circumference C pd or C 2pr

Area pr2

Postulate 1-9 If two figures are congruent, then their areas are equal .

Postulate 1-10 The area of a region is the sum of the areas of its non-overlapping parts.

Examples.

1 Finding Circumference G has a radius of 6.5 cm. Find the circumference of G in terms of . Then find the circumference to the nearest tenth.

C2 p r

( ) C 2 6.5

Formula for circumference of a circle Substitute 6.5 for r.

C 13

Exact answer

C 13

40.840704

Use a calculator.

The circumference of G is 13p , or about 40.8 cm.

6.5 cm G

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2 Finding Perimeter in the Coordinate Plane

y

Quadrilateral ABCD has vertices A(0, 0), B(9, 12),

12

C(11, 12), and D(2, 0). Find the perimeter.

10

Draw and label ABCD on a coordinate plane. 8

BC

6

4

2

A

D

x

O 2 4 6 8 10 12

Find the length of each side. Add the lengths to find the perimeter.

( ) ( ) AB % 9 0 2 12 0 2 %w9w2wwww12ww2

AB % 81 144 %ww2w25ww 15

BC u 11 9 u u 2 u 2

Use the Distance Formula. Ruler Postulate

( ) ( ) CD % 2 11 2 0 12 2 ( ) % 9 12 2

% 81 144 %ww2w25ww 15

DA u 2 0 u u 2 u 2

Use the Distance Formula. Ruler Postulate

Perimeter AB BC CD DA

Perimeter 15 2 15 2

Perimeter 34

The perimeter of quadrilateral ABCD is 34 units.

3 Finding Area of a Circle Find the area of B in terms of .

In B, r 1.5 yd.

A r2

( ) A 1.5 2

Formula for the area of a circle Substitute 1.5 for r.

A 2.25

The area of B is 2.25p yd 2.

B 1.5 yd

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Name_____________________________________ Class____________________________ Date ________________

10 ft

4 Finding Area of an Irregular Shape Find the area of the figure to the right.

Draw a horizontal line to separate the figure into three non-overlapping figures: one rectangle and two squares .

Find each area. Then add the areas.

AR bh

d Use the formulas. S AS s 2

( )( ) 15 5

d Substitute. S

( 5 )2

75

d Simplify. S

25

15 ft 5 ft

5 ft

5 ft

A 75 25 25 Add the areas.

A 125

Simplify.

The area of the figure is 125 ft2.

Quick Check. 1. a. Find the circumference of a circle with a radius of 18 m in terms of .

36p m

b. Find the circumference of a circle with a diameter of 18 m to the nearest tenth. 56.5 m

2. Graph quadrilateral KLMN with vertices K(3, 3), L(1, 3), M(1, 4), and N(3, 1). Find the perimeter of KLMN. 20 units

3. You are designing a rectangular banner for the front of a museum. The banner will be 4 ft wide and 7 yd high. How much material do you need in square yards? 913 yd2

y 4

M(1, 4)

N(3, 1) 2 O

4 2 2

K(3, 3)4

x 24

L(1, 3)

4. Copy the figure in Example 4. Separate it in a different way. Find the area. 125 ft2

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Geometry Lesson 1-9 33

10 ft 5 ft 5 ft

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