Study Guide and Intervention Points, Lines, and Planes
NAME ______________________________________________ DATE
____________ PERIOD _____
Reading to Learn Mathematics
1
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 1. As you study the chapter, complete each term’s definition or description. Remember to add the page number where you found the term. Add these pages to your Geometry Study Notebook to review vocabulary at the end of the chapter. Vocabulary Term
Found on Page
Definition/Description/Example
acute angle
adjacent angles uh·JAY·suhnt
angle
angle bisector
collinear koh·LIN·ee·uhr
complementary angles
congruent kuhn·GROO·uhnt
coplanar koh·PLAY·nuhr
line segment
linear pair
(continued on the next page) ©
Glencoe/McGraw-Hill
vii
Glencoe Geometry
Vocabulary Builder
Vocabulary Builder
NAME ______________________________________________ DATE
____________ PERIOD _____
Reading to Learn Mathematics
1
Vocabulary Builder Vocabulary Term
(continued)
Found on Page
Definition/Description/Example
midpoint
obtuse angle
perimeter
perpendicular lines
polygon PAHL·ee·gahn
ray
right angle
segment bisector
supplementary angles
vertical angles
©
Glencoe/McGraw-Hill
viii
Glencoe Geometry
NAME ______________________________________________ DATE
1-1
____________ PERIOD _____
Study Guide and Intervention Points, Lines, and Planes
Name Points, Lines, and Planes
In geometry, a point is a location, a line contains points, and a plane is a flat surface that contains points and lines. If points are on the same line, they are collinear. If points on are the same plane, they are coplanar. Use the figure to name each of the following.
!
a. a line containing point A
A
D B
C
N
The line can be named as !. Also, any two of the three points on the line can be used to name it. !"# AB , !"# AC , or !"# BC
Lesson 1-1
Example
b. a plane containing point D The plane can be named as plane N or can be named using three noncollinear points in the plane, such as plane ABD, plane ACD, and so on.
Exercises Refer to the figure.
A
!
1. Name a line that contains point A. m 2. What is another name for line
D B C E
P
m?
3. Name a point not on !"# AC . 4. Name the intersection of !"# AC and !"# DB . 5. Name a point not on line ! or line Draw and label a plane "# is in plane 6. !AB
m.
Q for each relationship. S
Q.
A
P T
"# at P. 7. !"# ST intersects !AB
Q
X
B Y
!
8. Point X is collinear with points A and P. 9. Point Y is not collinear with points T and P. 10. Line ! contains points X and Y.
©
Glencoe/McGraw-Hill
1
Glencoe Geometry
NAME ______________________________________________ DATE
1-1
____________ PERIOD _____
Study Guide and Intervention
(continued)
Points, Lines, and Planes Points, Lines, and Planes in Space all points. It contains lines and planes.
Space is a boundless, three-dimensional set of
Example a. How many planes appear in the figure? There are three planes: plane
O
P
N B
N , plane O, and plane P.
A
b. Are points A, B, and D coplanar? Yes. They are contained in plane
D
O.
C
Exercises Refer to the figure.
A
1. Name a line that is not contained in plane
N.
B C
2. Name a plane that contains point B.
N
D E
3. Name three collinear points.
Refer to the figure.
A
B
4. How many planes are shown in the figure? D
G
C
H I
5. Are points B, E, G, and H coplanar? Explain. F
E
J
6. Name a point coplanar with D, C, and E.
Draw and label a figure for each relationship. 7. Planes
8. Line r is in plane N , line s is in plane intersect at point J. 9. Line t contains point H and line plane N.
©
t
M andN intersect in !"# HJ .
Glencoe/McGraw-Hill
M , and lines r and s
N
M H
J
s r
t does not lie in plane M or
2
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
Skills Practice
1-1
Points, Lines, and Planes Refer to the figure.
A
1. Name a line that contains point D.
B
p
D
n
C
G
2. Name a point contained in line n.
4. Name the plane containing lines
Lesson 1-1
3. What is another name for line p ?
n and p.
Draw and label a figure for each relationship. 5. Point K lies on !"# RT . K
6. Plane
J contains line s.
T
R
s J
"# lies in plane B and contains 7. !YP point C, but does not contain point H. Y
C
8. Lines q and in plane U.
H
f
q
P
U
B
Refer to the figure.
f intersect at point Z
Z
F
9. How many planes are shown in the figure?
D E
A
10. How many of the planes contain points F and E?
C
W
B
11. Name four points that are coplanar.
12. Are points A, B, and C coplanar? Explain.
©
Glencoe/McGraw-Hill
3
Glencoe Geometry
NAME ______________________________________________ DATE
1-1
____________ PERIOD _____
Practice Points, Lines, and Planes
Refer to the figure.
j
1. Name a line that contains points T and P.
M
P
S
2. Name a line that intersects the plane containing points Q, N, and P.
T
R
Q N
h g
"#. 3. Name the plane that contains !"# TN and !QR Draw and label a figure for each relationship. "# and !CG "# intersect at point M 4. !AK in plane T. A
T
C
M
5. A line contains L(!4, !4) and M(2, 3). Line q is in the same coordinate plane but does "#. Line q contains point N. not intersect !LM y
G K
M
q x
O
L
N
Refer to the figure.
T
6. How many planes are shown in the figure?
W
7. Name three collinear points. A
8. Are points N, R, S, and W coplanar? Explain.
S X
M
Q P R N
VISUALIZATION Name the geometric term(s) modeled by each object. 9.
STOP
12. a car antenna
©
Glencoe/McGraw-Hill
10.
11.
tip of pin
strings
13. a library card
4
Glencoe Geometry
NAME ______________________________________________ DATE
1-1
____________ PERIOD _____
Reading to Learn Mathematics Points, Lines, and Planes
Pre-Activity
Why do chairs sometimes wobble? Read the introduction to Lesson 1-1 at the top of page 6 in your textbook.
• How many ways can you do this if you keep the pencil points in the same position? • How will your answer change if there are four pencil points?
Reading the Lesson 1. Complete each sentence. a. Points that lie on the same lie are called
points.
b. Points that do not lie in the same plane are called
points.
c. There is exactly one
through any two points.
d. There is exactly one
through any three noncollinear points.
2. Refer to the figure at the right. Indicate whether each statement is true or false.
D
U
a. Points A, B, and C are collinear. b. The intersection of plane ABC and line c. Line ! and line
C
m is point P.
!
B
P
A
m do not intersect.
m
d. Points A, P,and B can be used to name plane
U.
e. Line ! lies in plane ACB. 3. Complete the figure at the right to show the following relationship: Lines !, m, and n are coplanar and lie in plane Q. Lines ! and m intersect at point P. Line n intersects line m at R, but does not intersect line !.
Q
!
n
P R
m
Helping You Remember 4. Recall or look in a dictionary to find the meaning of the prefix co-. What does this prefix mean? How can it help you remember the meaning of collinear?
©
Glencoe/McGraw-Hill
5
Glencoe Geometry
Lesson 1-1
• Find three pencils of different lengths and hold them upright on your desk so that the three pencil points do not lie along a single line. Can you place a flat sheet of paper or cardboard so that it touches all three pencil points?
NAME ______________________________________________ DATE
1-1
____________ PERIOD _____
Enrichment
Points and Lines on a Matrix A matrix is a rectangular array of rows and columns. Points and lines on a matrix are not defined in the same way as in Euclidean geometry. A point on a matrix is a dot, which can be small or large. A line on a matrix is a path of dots that “line up.” Between two points on a line there may or may not be other points. Three examples of lines are shown at the upper right. The broad line can be thought of as a single line or as two narrow lines side by side. Dot-matrix printers for computers used dots to form characters. The dots are often called pixels. The matrix at the right shows how a dot-matrix printer might print the letter P.
Draw points on each matrix to create the given figures. 1. Draw two intersecting lines that have four points in common.
2. Draw two lines that cross but have no common points.
3. Make the number 0 (zero) so that it extends to the top and bottom sides of the matrix.
4. Make the capital letter O so that it extends to each side of the matrix.
5. Using separate grid paper, make dot designs for several other letters. Which were the easiest and which were the most difficult? ©
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6
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
Study Guide and Intervention
1-2
Linear Measure and Precision Measure Line Segments
A part of a line between two endpoints is called a line segment. The lengths of M !N ! and R !S ! are written as MN and RS. When you measure a segment, the precision of the measurement is half of the smallest unit on the ruler.
Example 1
Example 2
Find the length of M !N !.
M
N
cm
1
2
3
Find the length of R !S !.
R 4
S in.
The long marks are centimeters, and the shorter marks are millimeters. The length of !N M ! is 3.4 centimeters. The measurement is accurate to within 0.5 millimeter, so M !N ! is between 3.35 centimeters and 3.45 centimeters long.
1
2
The long marks are inches and the short marks are quarter inches. The length of R !S ! 3 4
is about 1"" inches. The measurement is accurate to within one half of a quarter inch, 1 8
5 8
!S ! is between 1"" inches and or "" inch, so R 7 8
Lesson 1-2
1"" inches long.
Exercises Find the length of each line segment or object. 1. A cm
2. S
B 1
2
3
T in.
3.
1
4. in.
1
2
cm
1
2
3
Find the precision for each measurement.
©
5. 10 in.
6. 32 mm
7. 44 cm
8. 2 ft
9. 3.5 mm
10. 2"" yd
Glencoe/McGraw-Hill
1 2
7
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
Study Guide and Intervention
1-2
(continued)
Linear Measure and Precision Calculate Measures
On !"# PQ, to say that point M is between points P and Q means P, Q, and M are collinear and PM # MQ $ PQ. On !"# AC, AB $ BC $ 3 cm. We can say that the segments are congruent, or A !B !"B !C !. Slashes on the figure indicate which segments are congruent.
Example 1 1.2 cm
1.9 cm
E
D
B
A
Example 2
Find EF.
M
P
Q C
Find x and AC.
2x ! 5
F
x
A
2x
B
C
Calculate EF by adding ED and DF.
B is between A and C.
ED # DF $ EF 1.2 # 1.9 $ EF 3.1 $ EF
AB # BC $ AC x # 2x $ 2x # 5 3x $ 2x # 5 x$5 AC $ 2x # 5 $ 2(5) # 5 $ 15
Therefore, E !F ! is 3.1 centimeters long.
Exercises Find the measurement of each segment. Assume that the art is not drawn to scale. 1. R !T !
2.0 cm
R
2. B !C !
2.5 cm
S
3. X !Z !
T
3 –21 in.
3 – 4
X
Y
in.
6 in.
A
2 –43
in. B
4. W !X !
6 cm
W
Z
C
X
Y
Find x and RS if S is between R and T. 5. RS $ 5x, ST $ 3x, and RT $ 48.
6. RS $ 2x, ST $ 5x # 4, and RT $ 32.
7. RS $ 6x, ST $12, and RT $ 72.
8. RS $ 4x, R !S !"S !T !, and RT $ 24.
Use the figures to determine whether each pair of segments is congruent. 9. A !B ! and C !D !
10. X !Y ! and Y !Z ! 11 cm
A 5 cm
B
X
D 5 cm
11 cm
3x ! 5
C
Y
©
Glencoe/McGraw-Hill
8
5x " 1
9x 2
Z
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
Skills Practice
1-2
Linear Measure and Precision Find the length of each line segment or object. 1.
2. cm
1
2
3
4
5 in.
1
2
Find the precision for each measurement. 1 2
5. 9"" inches
Lesson 1-2
4. 12 centimeters
3. 40 feet
Find the measurement of each segment. 6. N !Q !
7. A !C ! 1–41 in.
1in.
Q
P
8. G !H !
4.9 cm
A
N
5.2 cm
B
F
9.7 mm
C
G
H
15 mm
Find the value of the variable and YZ if Y is between X and Z. 9. XY $ 5p, YZ $ p, and XY $ 25
10. XY $ 12, YZ $ 2g, and XZ $ 28
11. XY $ 4m, YZ $ 3m, and XZ $ 42
12. XY $ 2c # 1, YZ $ 6c, and XZ $ 81
Use the figures to determine whether each pair of segments is congruent. 13. B !E !, C !D !
14. M !P !, N !P !
B 2m C 3m
E
©
3m 5m
D
Glencoe/McGraw-Hill
12 yd
M 12 yd
15. W !X !, W !Z ! P
Y
10 yd
5 ft
N
X
9
9 ft
Z 5 ft
W
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
Practice
1-2
Linear Measure and Precision Find the length of each line segment or object. 1. E
2.
F in.
1
2
cm
1
2
3
4
5
Find the precision for each measurement. 1 4
3. 120 meters
4. 7"" inches
5. 30.0 millimeters
Find the measurement of each segment. 6. P !S !
7. A !D ! 18.4 cm
P
2–83 in.
4.7 cm
Q
8. W !X !
S
A
W
1–41 in.
C
X
Y
89.6 cm 100 cm
D
Find the value of the variable and KL if K is between J and L. 9. JK $ 6r, KL $ 3r, and JL $ 27
10. JK $ 2s, KL $ s # 2, and JL $ 5s ! 10
Use the figures to determine whether each pair of segments is congruent. 11. T !U !, S !W !
12. A !D !, B !C !
T 2 ft S 2 ft
U
A
13. G !F !, F !E ! 12.7 in.
B
G
5x
3 ft 3 ft
W
H 6x
D
12.9 in.
C
14. CARPENTRY Jorge used the figure at the right to make a pattern for a mosaic he plans to inlay on a tabletop. Name all of the congruent segments in the figure.
F
E
A F
B
E
C D
©
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10
Glencoe Geometry
NAME ______________________________________________ DATE
1-2
____________ PERIOD _____
Reading to Learn Mathematics Linear Measure and Precision
Pre-Activity
Why are units of measure important? Read the introduction to Lesson 1-2 at the top of page 13 in your textbook. • The basic unit of length in the metric system is the meter. How many meters are there in one kilometer? • Do you think it would be easier to learn the relationships between the different units of length in the customary system (used in the United States) or in the metric system? Explain your answer.
Reading the Lesson
Lesson 1-2
1. Explain the difference between a line and a line segment and why one of these can be measured, while the other cannot.
2. What is the smallest length marked on a 12-inch ruler? What is the smallest length marked on a centimeter ruler? 3. Find the precision of each measurement. a. 15 cm b. 15.0 cm 4. Refer to the figure at the right. Which one of the following statements is true? Explain your answer. !B A !$C !D ! !B A !"C !D !
A 4.5 cm
D C
4.5 cm
B
5. Suppose that S is a point on V !W ! and S is not the same point as V or W. Tell whether each of the following statements is always, sometimes, or never true. a. VS $ SW b. S is between V and W. c. VS # VW $ SW
Helping You Remember 6. A good way to remember terms used in mathematics is to relate them to everyday words you know. Give three words that are used outside of mathematics that can help you remember that there are 100 centimeters in a meter.
©
Glencoe/McGraw-Hill
11
Glencoe Geometry
NAME ______________________________________________ DATE
1-2
____________ PERIOD _____
Enrichment
Points Equidistant from Segments The distance from a point to a segment is zero if the point is on the segment. Otherwise, it is the length of the shortest segment from the point to the segment. A figure is a locus if it is the set of all points that satisfy 1 4
a set of conditions. The locus of all points that are "" inch from the segment AB is shown by two dashed segments with semicircles at both ends.
A
B
1. Suppose A, B, C, and D are four different points, and consider the locus of all points x units from A !B ! and x units from C !D !. Use any unit you find convenient. The locus can take different forms. Sketch at least three possibilities. List some of the things that seem to affect the form of the locus. A C
B X
Y
R
B
D
A
Y A
X
C P
C
S D
B Q
D
2. Conduct your own investigation of the locus of points equidistant from two segments. Describe your results on a separate sheet of paper.
©
Glencoe/McGraw-Hill
12
Glencoe Geometry
NAME ______________________________________________ DATE
1-3
____________ PERIOD _____
Study Guide and Intervention Distance and Midpoints
Distance Between Two Points A
Pythagorean Theorem:
B
a
"5 "4 "3 "2 "1
d $ #! (x2 ! ! x1)2 #! (y2 !! y1)2
Find AB. B 0
1
B(1, 3)
Distance Formula:
AB $ | b ! a | or | a ! b |
A
y
a2 # b2 $ c2
b
Example 1
Distance in the Coordinate Plane
2
AB $ | (!4) ! 2 | $ |! 6 | $6
3
A(–2, –1)
x
O
C (1, –1)
Example 2
Find the distance between A(!2, !1) and B(1, 3). Pythagorean Theorem (AB)2 $ (AC)2 # (BC)2 (AB)2 $ (3)2 # (4)2 (AB)2 $ 25 AB $ #25 ! $5
Distance Formula d $ #! (x2 ! ! x1)2 #! (y2 !! y1)2 AB $ #! (1 ! (! !2))2 ! # (3 !! (!1))2! AB $ #! (3)2 #! (4)2 $ #25 ! $5
Exercises Use the number line to find each measure. 1. BD
2. DG
3. AF
4. EF
5. BG
6. AG
7. BE
8. DE
A
B
C
–10 –8 –6 –4 –2
DE 0
F 2
G 4
6
8
Use the Pythagorean Theorem to find the distance between each pair of points. 9. A(0, 0), B(6, 8) 11. M(1, !2), N(9, 13)
10. R(!2, 3), S(3, 15) 12. E(!12, 2), F(!9, 6)
Use the Distance Formula to find the distance between each pair of points. 13. A(0, 0), B(15, 20)
14. O(!12, 0), P(!8, 3)
15. C(11, !12), D(6, 2)
16. E(!2, 10), F(!4, 3)
©
Glencoe/McGraw-Hill
13
Glencoe Geometry
Lesson 1-3
Distance on a Number Line
NAME ______________________________________________ DATE
1-3
____________ PERIOD _____
Study Guide and Intervention
(continued)
Distance and Midpoints Midpoint of a Segment If the coordinates of the endpoints of a segment are a and b,
Midpoint on a Number Line
a #" b. then the coordinate of the midpoint of the segment is " 2
If a segment has endpoints with coordinates (x1, y1) and (x2, y2),
Midpoint on a Coordinate Plane
Example 1 P
$
x #x 2
y #y 2
%
1 2 1 2 " " then the coordinates of the midpoint of the segment are " ," .
Find the coordinate of the midpoint of P !Q !. Q
–3 –2 –1
0
1
2
The coordinates of P and Q are !3 and 1. !2 2
!3 # 1 2
!Q !, then the coordinate of M is "" $ "" or !1. If M is the midpoint of P
Example 2
M is the midpoint of P !Q ! for P("2, 4) and Q(4, 1). Find the coordinates of M.
$
x #x 2
y #y 2
% $ !22# 4
4#1 2
%
1 2 1 2 " " M$ " ," $ "", "" or (1, 2.5)
Exercises Use the number line to find the coordinate of the midpoint of each segment. 1. C !E !
2. D !G !
3. A !F !
4. E !G !
5. A !B !
6. B !G !
7. B !D !
8. D !E !
A
B
C
–10 –8 –6 –4 –2
D
EF
0
2
G 4
6
8
Find the coordinates of the midpoint of a segment having the given endpoints. 9. A(0, 0), B(12, 8)
10. R(!12, 8), S(6, 12)
11. M(11, !2), N(!9, 13)
12. E(!2, 6), F(!9, 3)
13. S(10, !22), T(9, 10)
14. M(!11, 2), N(!19, 6)
©
Glencoe/McGraw-Hill
14
Glencoe Geometry
NAME ______________________________________________ DATE
1-3
____________ PERIOD _____
Skills Practice Distance and Midpoints
Use the number line to find each measure. 1. LN
2. JL
3. KN
4. MN
J –6
K –4
L
–2
0
2
M 4
6
N 8
10
Use the Pythagorean Theorem to find the distance between each pair of points. 5.
6.
y
y
S
G O
x
O
x
F D
7. K(2, 3), F(4, 4)
8. C(!3, !1), Q(!2, 3)
Use the Distance Formula to find the distance between each pair of points. 10. W(!2, 2), R(5, 2)
11. A(!7, !3), B(5, 2)
Lesson 1-3
9. Y(2, 0), P(2, 6)
12. C(!3, 1), Q(2, 6)
Use the number line to find the coordinate of the midpoint of each segment. 13. D !E !
14. B !C !
15. B !D !
16. A !D !
A –6
–4
B –2
C 0
2
D 4
6
E 8
10
12
Find the coordinates of the midpoint of a segment having the given endpoints. 17. T(3, 1), U(5, 3)
18. J(!4, 2), F(5, !2)
Find the coordinates of the missing endpoint given that P is the midpoint of N !Q !. 19. N(2, 0), P(5, 2)
©
Glencoe/McGraw-Hill
20. N(5, 4), P(6, 3)
15
21. Q(3, 9), P(!1, 5)
Glencoe Geometry
NAME ______________________________________________ DATE
1-3
____________ PERIOD _____
Practice Distance and Midpoints
Use the number line to find each measure. 1. VW
2. TV
3. ST
4. SV
S –10
–8
–6
T
U
–4
–2
V 0
W 2
4
6
8
Use the Pythagorean Theorem to find the distance between each pair of points. 5.
6.
y
Z
M
O
y
S
O
x
x
E
Use the Distance Formula to find the distance between each pair of points. 8. U(1, 3), B(4, 6)
7. L(!7, 0), Y(5, 9)
Use the number line to find the coordinate of the midpoint of each segment. 9. R !T !
10. Q !R !
11. S !T !
12. P !R !
P –10
Q –8
–6
R –4
–2
S 0
T 2
4
6
Find the coordinates of the midpoint of a segment having the given endpoints. 13. K(!9, 3), H(5, 7)
14. W(!12, !7), T(!8, !4)
Find the coordinates of the missing endpoint given that E is the midpoint of D !F !. 15. F(5, 8), E(4, 3)
16. F(2, 9), E(!1, 6)
17. D(!3, !8), E(1, !2)
18. PERIMETER The coordinates of the vertices of a quadrilateral are R(!1, 3), S(3, 3), T(5, !1), and U(!2, !1). Find the perimeter of the quadrilateral. Round to the nearest tenth. ©
Glencoe/McGraw-Hill
16
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
Reading to Learn Mathematics
1-3
Distance and Midpoints Pre-Activity
How can you find the distance between two points without a ruler? Read the introduction to Lesson 1-3 at the top of page 21 in your textbook. • Look at the triangle in the introduction to this lesson. What is the special !B ! in this triangle? name for A • Find AB in this figure. Write your answer both as a radical and as a decimal number rounded to the nearest tenth.
Reading the Lesson 1. Match each formula or expression in the first column with one of the names in the second column. a. d $ #! (x2 ! ! x1)2 #! ( y2 !! y1)2
i. Pythagorean Theorem
a#b 2
b. ""
ii. Distance Formula in the Coordinate Plane
c. XY $ | a ! b |
iii. Midpoint of a Segment in the Coordinate Plane
d. c2 $ a2 # b2
iv. Distance Formula on a Number Line
$
x #x 2
y #y 2
1 2 1 2 ", "" e. "
%
v. Midpoint of a Segment on a Number Line
2. Fill in the steps to calculate the distance between the points M(4, !3) and N(!2, 7). ,
d$
#!!!! ( ! )2 # ( ! )2
MN $
#!!!! ( ! )2 # ( ! )2
MN $
#!! ( )2 # ( )2
MN $
#!! #
MN $
#!
).
Lesson 1-3
Let (x1, y1) $ (4, !3). Then (x2, y2) $ (
Find a decimal approximation for MN to the nearest hundredth.
Helping You Remember 3. A good way to remember a new formula in mathematics is to relate it to one you already know. If you forget the Distance Formula, how can you use the Pythagorean Theorem to find the distance d between two points on a coordinate plane?
©
Glencoe/McGraw-Hill
17
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
Enrichment
1-3
Lengths on a Grid Evenly-spaced horizontal and vertical lines form a grid. You can easily find segment lengths on a grid if the endpoints are grid-line intersections. For horizontal or vertical segments, simply count squares. For diagonal segments, use the Pythagorean Theorem (proven in Chapter 7). This theorem states that in any right triangle, if the length of the longest side (the side opposite the right angle) is c and the two shorter sides have lengths a and b, then c2 $ a2 # b2.
R
A C S D B I
Q E
Example
L
J
Find the measure of EF ! ! on the grid at the right. Locate a right triangle with E !F ! as its longest side.
F
K
N
M
E 2 5
EF $
#! 22 # 52!
F
$ #29 ! & 5.4 units
Find each measure to the nearest tenth of a unit. 1. !IJ !
2. M !! N
3. ! RS !
4. Q !S !
5. !I! K
6. J !! K
7. L !M !
8. L !! N
Use the grid above. Find the perimeter of each triangle to the nearest tenth of a unit. 9. ! ABC
10. !QRS
11. ! DEF
12. ! LMN
13. Of all the segments shown on the grid, which is longest? What is its length?
14. On the grid, 1 unit $ 0.5 cm. How can the answers above be used to find the measures in centimeters?
15. Use your answer from exercise 8 to calculate the length of segment LN in centimeters. Check by measuring with a centimeter ruler.
16. Use a centimeter ruler to find the perimeter of triangle IJK to the nearest tenth of a centimeter.
©
Glencoe/McGraw-Hill
18
Glencoe Geometry
____________ PERIOD _____
Reading to Learn Mathematics
1
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 1. As you study the chapter, complete each term’s definition or description. Remember to add the page number where you found the term. Add these pages to your Geometry Study Notebook to review vocabulary at the end of the chapter. Vocabulary Term
Found on Page
Definition/Description/Example
acute angle
adjacent angles uh·JAY·suhnt
angle
angle bisector
collinear koh·LIN·ee·uhr
complementary angles
congruent kuhn·GROO·uhnt
coplanar koh·PLAY·nuhr
line segment
linear pair
(continued on the next page) ©
Glencoe/McGraw-Hill
vii
Glencoe Geometry
Vocabulary Builder
Vocabulary Builder
NAME ______________________________________________ DATE
____________ PERIOD _____
Reading to Learn Mathematics
1
Vocabulary Builder Vocabulary Term
(continued)
Found on Page
Definition/Description/Example
midpoint
obtuse angle
perimeter
perpendicular lines
polygon PAHL·ee·gahn
ray
right angle
segment bisector
supplementary angles
vertical angles
©
Glencoe/McGraw-Hill
viii
Glencoe Geometry
NAME ______________________________________________ DATE
1-1
____________ PERIOD _____
Study Guide and Intervention Points, Lines, and Planes
Name Points, Lines, and Planes
In geometry, a point is a location, a line contains points, and a plane is a flat surface that contains points and lines. If points are on the same line, they are collinear. If points on are the same plane, they are coplanar. Use the figure to name each of the following.
!
a. a line containing point A
A
D B
C
N
The line can be named as !. Also, any two of the three points on the line can be used to name it. !"# AB , !"# AC , or !"# BC
Lesson 1-1
Example
b. a plane containing point D The plane can be named as plane N or can be named using three noncollinear points in the plane, such as plane ABD, plane ACD, and so on.
Exercises Refer to the figure.
A
!
1. Name a line that contains point A. m 2. What is another name for line
D B C E
P
m?
3. Name a point not on !"# AC . 4. Name the intersection of !"# AC and !"# DB . 5. Name a point not on line ! or line Draw and label a plane "# is in plane 6. !AB
m.
Q for each relationship. S
Q.
A
P T
"# at P. 7. !"# ST intersects !AB
Q
X
B Y
!
8. Point X is collinear with points A and P. 9. Point Y is not collinear with points T and P. 10. Line ! contains points X and Y.
©
Glencoe/McGraw-Hill
1
Glencoe Geometry
NAME ______________________________________________ DATE
1-1
____________ PERIOD _____
Study Guide and Intervention
(continued)
Points, Lines, and Planes Points, Lines, and Planes in Space all points. It contains lines and planes.
Space is a boundless, three-dimensional set of
Example a. How many planes appear in the figure? There are three planes: plane
O
P
N B
N , plane O, and plane P.
A
b. Are points A, B, and D coplanar? Yes. They are contained in plane
D
O.
C
Exercises Refer to the figure.
A
1. Name a line that is not contained in plane
N.
B C
2. Name a plane that contains point B.
N
D E
3. Name three collinear points.
Refer to the figure.
A
B
4. How many planes are shown in the figure? D
G
C
H I
5. Are points B, E, G, and H coplanar? Explain. F
E
J
6. Name a point coplanar with D, C, and E.
Draw and label a figure for each relationship. 7. Planes
8. Line r is in plane N , line s is in plane intersect at point J. 9. Line t contains point H and line plane N.
©
t
M andN intersect in !"# HJ .
Glencoe/McGraw-Hill
M , and lines r and s
N
M H
J
s r
t does not lie in plane M or
2
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
Skills Practice
1-1
Points, Lines, and Planes Refer to the figure.
A
1. Name a line that contains point D.
B
p
D
n
C
G
2. Name a point contained in line n.
4. Name the plane containing lines
Lesson 1-1
3. What is another name for line p ?
n and p.
Draw and label a figure for each relationship. 5. Point K lies on !"# RT . K
6. Plane
J contains line s.
T
R
s J
"# lies in plane B and contains 7. !YP point C, but does not contain point H. Y
C
8. Lines q and in plane U.
H
f
q
P
U
B
Refer to the figure.
f intersect at point Z
Z
F
9. How many planes are shown in the figure?
D E
A
10. How many of the planes contain points F and E?
C
W
B
11. Name four points that are coplanar.
12. Are points A, B, and C coplanar? Explain.
©
Glencoe/McGraw-Hill
3
Glencoe Geometry
NAME ______________________________________________ DATE
1-1
____________ PERIOD _____
Practice Points, Lines, and Planes
Refer to the figure.
j
1. Name a line that contains points T and P.
M
P
S
2. Name a line that intersects the plane containing points Q, N, and P.
T
R
Q N
h g
"#. 3. Name the plane that contains !"# TN and !QR Draw and label a figure for each relationship. "# and !CG "# intersect at point M 4. !AK in plane T. A
T
C
M
5. A line contains L(!4, !4) and M(2, 3). Line q is in the same coordinate plane but does "#. Line q contains point N. not intersect !LM y
G K
M
q x
O
L
N
Refer to the figure.
T
6. How many planes are shown in the figure?
W
7. Name three collinear points. A
8. Are points N, R, S, and W coplanar? Explain.
S X
M
Q P R N
VISUALIZATION Name the geometric term(s) modeled by each object. 9.
STOP
12. a car antenna
©
Glencoe/McGraw-Hill
10.
11.
tip of pin
strings
13. a library card
4
Glencoe Geometry
NAME ______________________________________________ DATE
1-1
____________ PERIOD _____
Reading to Learn Mathematics Points, Lines, and Planes
Pre-Activity
Why do chairs sometimes wobble? Read the introduction to Lesson 1-1 at the top of page 6 in your textbook.
• How many ways can you do this if you keep the pencil points in the same position? • How will your answer change if there are four pencil points?
Reading the Lesson 1. Complete each sentence. a. Points that lie on the same lie are called
points.
b. Points that do not lie in the same plane are called
points.
c. There is exactly one
through any two points.
d. There is exactly one
through any three noncollinear points.
2. Refer to the figure at the right. Indicate whether each statement is true or false.
D
U
a. Points A, B, and C are collinear. b. The intersection of plane ABC and line c. Line ! and line
C
m is point P.
!
B
P
A
m do not intersect.
m
d. Points A, P,and B can be used to name plane
U.
e. Line ! lies in plane ACB. 3. Complete the figure at the right to show the following relationship: Lines !, m, and n are coplanar and lie in plane Q. Lines ! and m intersect at point P. Line n intersects line m at R, but does not intersect line !.
Q
!
n
P R
m
Helping You Remember 4. Recall or look in a dictionary to find the meaning of the prefix co-. What does this prefix mean? How can it help you remember the meaning of collinear?
©
Glencoe/McGraw-Hill
5
Glencoe Geometry
Lesson 1-1
• Find three pencils of different lengths and hold them upright on your desk so that the three pencil points do not lie along a single line. Can you place a flat sheet of paper or cardboard so that it touches all three pencil points?
NAME ______________________________________________ DATE
1-1
____________ PERIOD _____
Enrichment
Points and Lines on a Matrix A matrix is a rectangular array of rows and columns. Points and lines on a matrix are not defined in the same way as in Euclidean geometry. A point on a matrix is a dot, which can be small or large. A line on a matrix is a path of dots that “line up.” Between two points on a line there may or may not be other points. Three examples of lines are shown at the upper right. The broad line can be thought of as a single line or as two narrow lines side by side. Dot-matrix printers for computers used dots to form characters. The dots are often called pixels. The matrix at the right shows how a dot-matrix printer might print the letter P.
Draw points on each matrix to create the given figures. 1. Draw two intersecting lines that have four points in common.
2. Draw two lines that cross but have no common points.
3. Make the number 0 (zero) so that it extends to the top and bottom sides of the matrix.
4. Make the capital letter O so that it extends to each side of the matrix.
5. Using separate grid paper, make dot designs for several other letters. Which were the easiest and which were the most difficult? ©
Glencoe/McGraw-Hill
6
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
Study Guide and Intervention
1-2
Linear Measure and Precision Measure Line Segments
A part of a line between two endpoints is called a line segment. The lengths of M !N ! and R !S ! are written as MN and RS. When you measure a segment, the precision of the measurement is half of the smallest unit on the ruler.
Example 1
Example 2
Find the length of M !N !.
M
N
cm
1
2
3
Find the length of R !S !.
R 4
S in.
The long marks are centimeters, and the shorter marks are millimeters. The length of !N M ! is 3.4 centimeters. The measurement is accurate to within 0.5 millimeter, so M !N ! is between 3.35 centimeters and 3.45 centimeters long.
1
2
The long marks are inches and the short marks are quarter inches. The length of R !S ! 3 4
is about 1"" inches. The measurement is accurate to within one half of a quarter inch, 1 8
5 8
!S ! is between 1"" inches and or "" inch, so R 7 8
Lesson 1-2
1"" inches long.
Exercises Find the length of each line segment or object. 1. A cm
2. S
B 1
2
3
T in.
3.
1
4. in.
1
2
cm
1
2
3
Find the precision for each measurement.
©
5. 10 in.
6. 32 mm
7. 44 cm
8. 2 ft
9. 3.5 mm
10. 2"" yd
Glencoe/McGraw-Hill
1 2
7
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
Study Guide and Intervention
1-2
(continued)
Linear Measure and Precision Calculate Measures
On !"# PQ, to say that point M is between points P and Q means P, Q, and M are collinear and PM # MQ $ PQ. On !"# AC, AB $ BC $ 3 cm. We can say that the segments are congruent, or A !B !"B !C !. Slashes on the figure indicate which segments are congruent.
Example 1 1.2 cm
1.9 cm
E
D
B
A
Example 2
Find EF.
M
P
Q C
Find x and AC.
2x ! 5
F
x
A
2x
B
C
Calculate EF by adding ED and DF.
B is between A and C.
ED # DF $ EF 1.2 # 1.9 $ EF 3.1 $ EF
AB # BC $ AC x # 2x $ 2x # 5 3x $ 2x # 5 x$5 AC $ 2x # 5 $ 2(5) # 5 $ 15
Therefore, E !F ! is 3.1 centimeters long.
Exercises Find the measurement of each segment. Assume that the art is not drawn to scale. 1. R !T !
2.0 cm
R
2. B !C !
2.5 cm
S
3. X !Z !
T
3 –21 in.
3 – 4
X
Y
in.
6 in.
A
2 –43
in. B
4. W !X !
6 cm
W
Z
C
X
Y
Find x and RS if S is between R and T. 5. RS $ 5x, ST $ 3x, and RT $ 48.
6. RS $ 2x, ST $ 5x # 4, and RT $ 32.
7. RS $ 6x, ST $12, and RT $ 72.
8. RS $ 4x, R !S !"S !T !, and RT $ 24.
Use the figures to determine whether each pair of segments is congruent. 9. A !B ! and C !D !
10. X !Y ! and Y !Z ! 11 cm
A 5 cm
B
X
D 5 cm
11 cm
3x ! 5
C
Y
©
Glencoe/McGraw-Hill
8
5x " 1
9x 2
Z
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
Skills Practice
1-2
Linear Measure and Precision Find the length of each line segment or object. 1.
2. cm
1
2
3
4
5 in.
1
2
Find the precision for each measurement. 1 2
5. 9"" inches
Lesson 1-2
4. 12 centimeters
3. 40 feet
Find the measurement of each segment. 6. N !Q !
7. A !C ! 1–41 in.
1in.
Q
P
8. G !H !
4.9 cm
A
N
5.2 cm
B
F
9.7 mm
C
G
H
15 mm
Find the value of the variable and YZ if Y is between X and Z. 9. XY $ 5p, YZ $ p, and XY $ 25
10. XY $ 12, YZ $ 2g, and XZ $ 28
11. XY $ 4m, YZ $ 3m, and XZ $ 42
12. XY $ 2c # 1, YZ $ 6c, and XZ $ 81
Use the figures to determine whether each pair of segments is congruent. 13. B !E !, C !D !
14. M !P !, N !P !
B 2m C 3m
E
©
3m 5m
D
Glencoe/McGraw-Hill
12 yd
M 12 yd
15. W !X !, W !Z ! P
Y
10 yd
5 ft
N
X
9
9 ft
Z 5 ft
W
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
Practice
1-2
Linear Measure and Precision Find the length of each line segment or object. 1. E
2.
F in.
1
2
cm
1
2
3
4
5
Find the precision for each measurement. 1 4
3. 120 meters
4. 7"" inches
5. 30.0 millimeters
Find the measurement of each segment. 6. P !S !
7. A !D ! 18.4 cm
P
2–83 in.
4.7 cm
Q
8. W !X !
S
A
W
1–41 in.
C
X
Y
89.6 cm 100 cm
D
Find the value of the variable and KL if K is between J and L. 9. JK $ 6r, KL $ 3r, and JL $ 27
10. JK $ 2s, KL $ s # 2, and JL $ 5s ! 10
Use the figures to determine whether each pair of segments is congruent. 11. T !U !, S !W !
12. A !D !, B !C !
T 2 ft S 2 ft
U
A
13. G !F !, F !E ! 12.7 in.
B
G
5x
3 ft 3 ft
W
H 6x
D
12.9 in.
C
14. CARPENTRY Jorge used the figure at the right to make a pattern for a mosaic he plans to inlay on a tabletop. Name all of the congruent segments in the figure.
F
E
A F
B
E
C D
©
Glencoe/McGraw-Hill
10
Glencoe Geometry
NAME ______________________________________________ DATE
1-2
____________ PERIOD _____
Reading to Learn Mathematics Linear Measure and Precision
Pre-Activity
Why are units of measure important? Read the introduction to Lesson 1-2 at the top of page 13 in your textbook. • The basic unit of length in the metric system is the meter. How many meters are there in one kilometer? • Do you think it would be easier to learn the relationships between the different units of length in the customary system (used in the United States) or in the metric system? Explain your answer.
Reading the Lesson
Lesson 1-2
1. Explain the difference between a line and a line segment and why one of these can be measured, while the other cannot.
2. What is the smallest length marked on a 12-inch ruler? What is the smallest length marked on a centimeter ruler? 3. Find the precision of each measurement. a. 15 cm b. 15.0 cm 4. Refer to the figure at the right. Which one of the following statements is true? Explain your answer. !B A !$C !D ! !B A !"C !D !
A 4.5 cm
D C
4.5 cm
B
5. Suppose that S is a point on V !W ! and S is not the same point as V or W. Tell whether each of the following statements is always, sometimes, or never true. a. VS $ SW b. S is between V and W. c. VS # VW $ SW
Helping You Remember 6. A good way to remember terms used in mathematics is to relate them to everyday words you know. Give three words that are used outside of mathematics that can help you remember that there are 100 centimeters in a meter.
©
Glencoe/McGraw-Hill
11
Glencoe Geometry
NAME ______________________________________________ DATE
1-2
____________ PERIOD _____
Enrichment
Points Equidistant from Segments The distance from a point to a segment is zero if the point is on the segment. Otherwise, it is the length of the shortest segment from the point to the segment. A figure is a locus if it is the set of all points that satisfy 1 4
a set of conditions. The locus of all points that are "" inch from the segment AB is shown by two dashed segments with semicircles at both ends.
A
B
1. Suppose A, B, C, and D are four different points, and consider the locus of all points x units from A !B ! and x units from C !D !. Use any unit you find convenient. The locus can take different forms. Sketch at least three possibilities. List some of the things that seem to affect the form of the locus. A C
B X
Y
R
B
D
A
Y A
X
C P
C
S D
B Q
D
2. Conduct your own investigation of the locus of points equidistant from two segments. Describe your results on a separate sheet of paper.
©
Glencoe/McGraw-Hill
12
Glencoe Geometry
NAME ______________________________________________ DATE
1-3
____________ PERIOD _____
Study Guide and Intervention Distance and Midpoints
Distance Between Two Points A
Pythagorean Theorem:
B
a
"5 "4 "3 "2 "1
d $ #! (x2 ! ! x1)2 #! (y2 !! y1)2
Find AB. B 0
1
B(1, 3)
Distance Formula:
AB $ | b ! a | or | a ! b |
A
y
a2 # b2 $ c2
b
Example 1
Distance in the Coordinate Plane
2
AB $ | (!4) ! 2 | $ |! 6 | $6
3
A(–2, –1)
x
O
C (1, –1)
Example 2
Find the distance between A(!2, !1) and B(1, 3). Pythagorean Theorem (AB)2 $ (AC)2 # (BC)2 (AB)2 $ (3)2 # (4)2 (AB)2 $ 25 AB $ #25 ! $5
Distance Formula d $ #! (x2 ! ! x1)2 #! (y2 !! y1)2 AB $ #! (1 ! (! !2))2 ! # (3 !! (!1))2! AB $ #! (3)2 #! (4)2 $ #25 ! $5
Exercises Use the number line to find each measure. 1. BD
2. DG
3. AF
4. EF
5. BG
6. AG
7. BE
8. DE
A
B
C
–10 –8 –6 –4 –2
DE 0
F 2
G 4
6
8
Use the Pythagorean Theorem to find the distance between each pair of points. 9. A(0, 0), B(6, 8) 11. M(1, !2), N(9, 13)
10. R(!2, 3), S(3, 15) 12. E(!12, 2), F(!9, 6)
Use the Distance Formula to find the distance between each pair of points. 13. A(0, 0), B(15, 20)
14. O(!12, 0), P(!8, 3)
15. C(11, !12), D(6, 2)
16. E(!2, 10), F(!4, 3)
©
Glencoe/McGraw-Hill
13
Glencoe Geometry
Lesson 1-3
Distance on a Number Line
NAME ______________________________________________ DATE
1-3
____________ PERIOD _____
Study Guide and Intervention
(continued)
Distance and Midpoints Midpoint of a Segment If the coordinates of the endpoints of a segment are a and b,
Midpoint on a Number Line
a #" b. then the coordinate of the midpoint of the segment is " 2
If a segment has endpoints with coordinates (x1, y1) and (x2, y2),
Midpoint on a Coordinate Plane
Example 1 P
$
x #x 2
y #y 2
%
1 2 1 2 " " then the coordinates of the midpoint of the segment are " ," .
Find the coordinate of the midpoint of P !Q !. Q
–3 –2 –1
0
1
2
The coordinates of P and Q are !3 and 1. !2 2
!3 # 1 2
!Q !, then the coordinate of M is "" $ "" or !1. If M is the midpoint of P
Example 2
M is the midpoint of P !Q ! for P("2, 4) and Q(4, 1). Find the coordinates of M.
$
x #x 2
y #y 2
% $ !22# 4
4#1 2
%
1 2 1 2 " " M$ " ," $ "", "" or (1, 2.5)
Exercises Use the number line to find the coordinate of the midpoint of each segment. 1. C !E !
2. D !G !
3. A !F !
4. E !G !
5. A !B !
6. B !G !
7. B !D !
8. D !E !
A
B
C
–10 –8 –6 –4 –2
D
EF
0
2
G 4
6
8
Find the coordinates of the midpoint of a segment having the given endpoints. 9. A(0, 0), B(12, 8)
10. R(!12, 8), S(6, 12)
11. M(11, !2), N(!9, 13)
12. E(!2, 6), F(!9, 3)
13. S(10, !22), T(9, 10)
14. M(!11, 2), N(!19, 6)
©
Glencoe/McGraw-Hill
14
Glencoe Geometry
NAME ______________________________________________ DATE
1-3
____________ PERIOD _____
Skills Practice Distance and Midpoints
Use the number line to find each measure. 1. LN
2. JL
3. KN
4. MN
J –6
K –4
L
–2
0
2
M 4
6
N 8
10
Use the Pythagorean Theorem to find the distance between each pair of points. 5.
6.
y
y
S
G O
x
O
x
F D
7. K(2, 3), F(4, 4)
8. C(!3, !1), Q(!2, 3)
Use the Distance Formula to find the distance between each pair of points. 10. W(!2, 2), R(5, 2)
11. A(!7, !3), B(5, 2)
Lesson 1-3
9. Y(2, 0), P(2, 6)
12. C(!3, 1), Q(2, 6)
Use the number line to find the coordinate of the midpoint of each segment. 13. D !E !
14. B !C !
15. B !D !
16. A !D !
A –6
–4
B –2
C 0
2
D 4
6
E 8
10
12
Find the coordinates of the midpoint of a segment having the given endpoints. 17. T(3, 1), U(5, 3)
18. J(!4, 2), F(5, !2)
Find the coordinates of the missing endpoint given that P is the midpoint of N !Q !. 19. N(2, 0), P(5, 2)
©
Glencoe/McGraw-Hill
20. N(5, 4), P(6, 3)
15
21. Q(3, 9), P(!1, 5)
Glencoe Geometry
NAME ______________________________________________ DATE
1-3
____________ PERIOD _____
Practice Distance and Midpoints
Use the number line to find each measure. 1. VW
2. TV
3. ST
4. SV
S –10
–8
–6
T
U
–4
–2
V 0
W 2
4
6
8
Use the Pythagorean Theorem to find the distance between each pair of points. 5.
6.
y
Z
M
O
y
S
O
x
x
E
Use the Distance Formula to find the distance between each pair of points. 8. U(1, 3), B(4, 6)
7. L(!7, 0), Y(5, 9)
Use the number line to find the coordinate of the midpoint of each segment. 9. R !T !
10. Q !R !
11. S !T !
12. P !R !
P –10
Q –8
–6
R –4
–2
S 0
T 2
4
6
Find the coordinates of the midpoint of a segment having the given endpoints. 13. K(!9, 3), H(5, 7)
14. W(!12, !7), T(!8, !4)
Find the coordinates of the missing endpoint given that E is the midpoint of D !F !. 15. F(5, 8), E(4, 3)
16. F(2, 9), E(!1, 6)
17. D(!3, !8), E(1, !2)
18. PERIMETER The coordinates of the vertices of a quadrilateral are R(!1, 3), S(3, 3), T(5, !1), and U(!2, !1). Find the perimeter of the quadrilateral. Round to the nearest tenth. ©
Glencoe/McGraw-Hill
16
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
Reading to Learn Mathematics
1-3
Distance and Midpoints Pre-Activity
How can you find the distance between two points without a ruler? Read the introduction to Lesson 1-3 at the top of page 21 in your textbook. • Look at the triangle in the introduction to this lesson. What is the special !B ! in this triangle? name for A • Find AB in this figure. Write your answer both as a radical and as a decimal number rounded to the nearest tenth.
Reading the Lesson 1. Match each formula or expression in the first column with one of the names in the second column. a. d $ #! (x2 ! ! x1)2 #! ( y2 !! y1)2
i. Pythagorean Theorem
a#b 2
b. ""
ii. Distance Formula in the Coordinate Plane
c. XY $ | a ! b |
iii. Midpoint of a Segment in the Coordinate Plane
d. c2 $ a2 # b2
iv. Distance Formula on a Number Line
$
x #x 2
y #y 2
1 2 1 2 ", "" e. "
%
v. Midpoint of a Segment on a Number Line
2. Fill in the steps to calculate the distance between the points M(4, !3) and N(!2, 7). ,
d$
#!!!! ( ! )2 # ( ! )2
MN $
#!!!! ( ! )2 # ( ! )2
MN $
#!! ( )2 # ( )2
MN $
#!! #
MN $
#!
).
Lesson 1-3
Let (x1, y1) $ (4, !3). Then (x2, y2) $ (
Find a decimal approximation for MN to the nearest hundredth.
Helping You Remember 3. A good way to remember a new formula in mathematics is to relate it to one you already know. If you forget the Distance Formula, how can you use the Pythagorean Theorem to find the distance d between two points on a coordinate plane?
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Glencoe/McGraw-Hill
17
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
Enrichment
1-3
Lengths on a Grid Evenly-spaced horizontal and vertical lines form a grid. You can easily find segment lengths on a grid if the endpoints are grid-line intersections. For horizontal or vertical segments, simply count squares. For diagonal segments, use the Pythagorean Theorem (proven in Chapter 7). This theorem states that in any right triangle, if the length of the longest side (the side opposite the right angle) is c and the two shorter sides have lengths a and b, then c2 $ a2 # b2.
R
A C S D B I
Q E
Example
L
J
Find the measure of EF ! ! on the grid at the right. Locate a right triangle with E !F ! as its longest side.
F
K
N
M
E 2 5
EF $
#! 22 # 52!
F
$ #29 ! & 5.4 units
Find each measure to the nearest tenth of a unit. 1. !IJ !
2. M !! N
3. ! RS !
4. Q !S !
5. !I! K
6. J !! K
7. L !M !
8. L !! N
Use the grid above. Find the perimeter of each triangle to the nearest tenth of a unit. 9. ! ABC
10. !QRS
11. ! DEF
12. ! LMN
13. Of all the segments shown on the grid, which is longest? What is its length?
14. On the grid, 1 unit $ 0.5 cm. How can the answers above be used to find the measures in centimeters?
15. Use your answer from exercise 8 to calculate the length of segment LN in centimeters. Check by measuring with a centimeter ruler.
16. Use a centimeter ruler to find the perimeter of triangle IJK to the nearest tenth of a centimeter.
©
Glencoe/McGraw-Hill
18
Glencoe Geometry
