Download CH 3 Review

NAME
~
DATE
~------------------
Practice with Exa.mples
For use with pages 129-134
Identify relationships between lines and identify angles
formed by transversals
VOCABULARY
Two lines are parallel lines if they are coplanar and do not intersect.
Lines that do not intersect and are not coplanar are called skew lines.
Two planes that do not intersect are called parallel planes.
I
A transversal is a line that intersects two or more coplanar lines at
different points.
When two lines are cut by a transversal, two angles are corresponding
angles if they occupy corresponding positions.
I
When two lines are cut by a transversal, two angles are alternate
exterior angles if they lie outside the two lines on opposite sides of the
transversal.
When two lines are cut by a transversal, two angles are alternate
interior angles if they lie between the two lines on opposite sides of-the
transversal.
,
When two lines are cut by a transversal, two angles are consecutive
interior angles (or same side interior angles) if they lie between the
two lines on the same side of the transversal .
. Postulate 13 Parallel Postulate If there is a line and a point not on the
line, then there is exactly one line through the point parallel to the given
line.
Postulate 14 Perpendicular Postulate If there is a line and a point not
on the line, then there is exactly one line through the point perpendicular
I to the given line.
G1IfI)
Identifyi~g Relationships in Space
Think of each segment in the diagram as part of a line.
Which of the lines ~ppear to fit the description?
~
~
a. parallel to AB
b. skew to AB
H
c. parallel to Be
d. Are planes ABE and CDE
parallel?
E
A
c
B
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Geometry
Practice Workbook
vvith Examples
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LESSON
3.1
CONTINUED
NAME
~
_
DATE
Practice with Examples
For use with pages 129--134
SOLUTION
~
~
a. Only CD is parallel to AB.
~
~
~
b. ED and EC are skew to AB.
~
~
c. Only AD is parallel to Be.
d. No, the two planes are not parallel. At the very least, we can see that
the two planes intersect at point E .
.~~f!!.~~~I!.~.
!l!.~.~lf.i!.'!!l!.~'!.
.!
: .
Think of each segment in the diagram as part of a line.
Fill in the blank with parallel, skew, or perpendicular.
~
~
1. DE and CFare ------A
~~
~
2. AD, BE, and CF are ------3. Plane ABC and plane DEF are ~
B
o
_
~
~
4. BE and AB are -------
Think of each segment in the diagram as part of a line.
There may be more than one right answer.
5. Name a line perpendicular to
liD.
B
c
6. Name a plane parallel to DCH.
G
H
.~
'fl
.
~
7. Name a lme parallel to Be.
I
II,
Iii
.
~
8. Name a line skew to FG.
Geometry
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Practice Workbook
with Examples
LESSON
3.1
NAME'
_
DATE
Practice with Examples
CONTINUED
For use with pages 129-134
List all pairs of angles that fit the description.
a. corresponding
b. alternate exterior
c. alternate interior
d. consecutive interior
a. Ll
L2
L8
L7
and
and
and
and
L3
L4
L6
L5
b. Ll and L5
L8 and L4
c. L2 and L6
L7 and L3
d. L2 and L3
L7 and L6
Exercises for Example 2
••••••e •••••••••••••••••••••••••••••••••••••••• e ~ ••••••••••••••••••••••••
~ •••.••••••••~ ••••••••••••••••••••" ••••••••••" •••••••••.•••••••••••••••••••••••••••••.••••••••••••
e! ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
e ••••••••
Complete the statement with corresponding, alternate interior,
alternate exterior, or consecutive interior.
9. L4 and L8 are
angles.
10. L2 and L6 are
angles.
11. Ll and L8 are
angles.
4 1
12. L8 and L2 are
angles:
13. L4 and L5 are ~
angles.
14. L5 and L 1 are
angles.
Geometll'Y
Practice Workbook vvith Examples
3 2
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NAME
_
DATE
Practice with Examples
For use with pages 136-141
Write different types of proofs and prove results
about perpendicular lines
VOCABULARY
A flow proof uses arrows to show the flow of the logical argument.
:
.
,
,.
,
i
I
Ii
'Q I
ii,
I
Theorem 3.1 If two lines intersect to form a linear pair of congruent
angles, then the lines are perpendicular .
Theorem 3.2 If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
Theorem 3.3 If two lines are perpendicular, then they intersect to form
four right angles.
I
I;,
,
Comparing Types of Proofs
Write a two-column proof of Theorem 3.1 (a flow proof
is provided in Example 2 on page 137 of ~he text).
9
Given:
Ll
=: L2,
Ll and L2 are a linear pair.
2
Prove:
g.lh
h
Reasons
Statements
1. Ll
2. mLl
3.
4.
5.
6.
7.
,
,
8.
9.
[':
10.
"
I
(,
!
=: L2
=
1. Given
mL2
2. Definition of congruent angles
L 1 and L2 are a linear pair
L 1 and L2 are supplementary
mL1 + mL2 = 180
mL1 + mLl = 180
2 . (mLl) = 180
mL1 = 90
L 1 is a right L
g.l h
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0
0
0
0
3.
4.
5.
6.
7.
8.
9.
10.
Given
Linear Pair Postulate
Definition of supplementary angles
Substitution property of equality
Distributive property
Division property of equality
Definition of right angle
Definition of perpendicular lines
Geometry
Practice Workbook
vvith Examples
,
L~SSON
NAME
DATE -
~~------------------------
,,--_
Practice with Examples
CONTINUED
For use with pages 136-141
,
Exercises/or
Example 1
:
.
1. Write a two-column proof of Theorem 3.2. Note that you are asked to complete
a paragraph proof of this theorem in Practice and Applications Exercise 17 on
page 139.
2. Write a paragraph proof of Theorem 3.3. Note that you are asked to complete a flow
proof related to this theorem in Practice and Applications Exercise 18 on page 139
and a two-column proof related to this theorem in Exercise 19.
mtI!D
.tl
Application of the Theorems
Find the value of x.
b.
a.
m
n
$OUJTION.
a.x
= 90 because, by Theorem 3.3, since k and
e are perpendicular,
all four angles
formed are right angles. By definition of a right angle, x is 90~
b. By Theorem 3.3, since m and n are perpendicular, all four angles formed are right
angles. By Theorem 3.2, the 62° angle and the X angle are complementary. Thus
x + 62 = 90, so x = 28.
.
O
Geometry
Practice Workbook with Examples
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LESSON
3.2
CONTINUED
NAME
~
_
DATE
Practice with Examples
For use with pages 136-141
Exercises for Example 2
'.
.
•••••••••
~••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
'! •••••••••••••••••••••••••
'.'•••••••••••••••
" ••••••••••••••••••
Find the value of x.
1.
2.
3.
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Geometry
Practice Workbook
with Examples
/.
NAME~
__ ~
~
~
_ DATE
Practice with Examples
For use with pages 143-149
Prove and use results about parallel lines and transversals
properties of parallel lines to solve problems
I
and use
VOCABULARY
Postulate 15 Corresponding Angles Postulate If two parallel lines are cut
by a transversal, then the pairs of corresponding angles are congruent.
Theorem 3.4 If two parallel lines are cut by a transversal, then the pairs
.of alternate .interior angles are congruent.
Theorem 3.5 If two parallel lines are cut by a transversal, then the pairs
of consecutive interior angles are supplementary.
Theorem 3.6 If two parallel lines are cut by a transversal, then the pairs
of alternate exterior angles are congruent.
Theorem 3.7 If a transversal is perpendicular to one of two parallel
lines, then it is perpendicular to the other.
Given that mL 1 = 32°, find each measure. Tell which postulate or
theorem you use.
(
\--.:.
a. mL2
b. mL3
c. mL4
d. mLS
SOLUTmON
a. mL2 = 32°
Corresponding Angles Postulate
b. mL3 = 32°
c. mL4
= 180° - mL3=
d. mLS = 32°
Geometry
Practice Workbook with Examples
Alternate Exterior Angles Theorem
148
0
Linear Pair Postulate
Vertical Angles Theorem
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'!
LESSON
CONTINUED
NAME
~
_
DATE
Practice with Examples
For use with pages 143~149
.~~.~r.~~~I!.~.
!.f!.~.~l!.l!.'!!l!.~~.!
Find each measure given that mL6
1. mL7
2. mL8
3. mL9
4. mLIO
5. mLll
6. ntL12
.
=
67
0
•
7. mL13
__
!!..sing Properties of Parallel Lines
Use properties of parallel lines to find the value of x.
(x - 8)° ~ 55°
x
=
63°
Alternate Exterior Angles Theorem
Add,
Exercises for Example 2
........................................................................................................................................
Use properties of parallel lines to find the value of x.
8.
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9.
Geometry
Practice Workbook
with Examples
LESSON
3.3
NAME
~----~------
~
_
DATE
Practice with Examples
CONTINUED
For use with pages 143-149
10.
11.
i
98°
(lOx
-
I
\
2)°
!
I
rI
f
12.
I
I
13.
t
f
I
'f
'I
"
14.
15.
Geometry
Practice Workbook
with Examples
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NAME
~
_
DATE
Practice lII.1ith Examples
For use with pages 150-156
'.
Prove that two lines are parallel and use properties of parallel lines to
solve problems
.
'VOCABULARV
Postulate 16 Corresponding Angles Converse If two lines are cut by a
transversal so that corresponding angles are congruent, then the lines are
parallel.
~-
!
I
Theorem 3.8 Alternate Interior Angles Converse If two lines are cut
by a transversal so that alternate interior angles are congruent, then the
lines are parallel.
!
I,
i
.Il'>
I·
Theorem 3.9 Consecutive Interior Angles Converse If two lines are
cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel.
Theorem 3.10 Alternate Exterior Angles Converse If two lines are cut
by a transversal so that alternate exterior angles are congruent, then the
lines are parallel.
Proving that Two Lines are Parallel
Prove that lines j and k are parallel.
~--~..----~
j
__
Given: mL 1
mL2
Prove: j
=
53
=
127
~~
k
0
0
II k
Statements
Reasons
= 53
0
1. mLl
1. Given
mL2 = 127
0
2. mL3
+ mL2 =
180
3. mL3 + 127 = 180
53
5. mL3 = mLI
0
0
4. mL3=
6. L3
7. -j
= Ll
II k
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0
0
2. Linear Pair Postulate
3. Substitution prop. of equality
. 4. Subtraction prop. of equality
5. Substitution prop. of equality
6. Def. of congruent angles
7. Corresponding Angles Converse
Geometry
. Practice Workbook
vvith Examples
i
!.~
·1
LESSON
NAME
3.4
CONTINUED
~
~--------~--
__
i
DATE -'-
_
Practice with Examples
For use with pages 150-156
{~.r:!.~~~l!.f!.
!.l!.~.~1!.i!.'!!.I?~i!..!
;
:
,.............•..............
Prove the statement from the given information.
1. Prove:
e 1/ m
2. Prove: n II
0
Identifying Parallel Lines
Determine which rays are. parallel.
---7
---7
a. Is PN parallel to SR?
---7
(
---7
b. Is PO parallel to SQ?
SOLUTION
---7
a. Decide whether PN
mLNPS
=
---7
II SR.
39° + 101°
= 140°
mLRSP
=
42°
+
98°
= 140°
---7
---7
LNPS and LRSP are congruent alternate interior angles, so PN II SR.
---7
b. Decide whether PO
mLOPS
= 101°
mLPSQ
=
-7
1/
SQ.
98°
LOPS and LPSQ are alternate interior angles, but they are not
---7
~
.
congruent, so PO and S!,L are not parallel.
Geometry
Practice
Workbook with Examples
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LESSON
3.4
NAME
DATE
~--------------------------
Practice with Examples
CONTINUED
For use with pages 150-156
~~~!.t?~~~.~.
!.l!!.
.~l!.;!.T!!l!.~f!.
.?
Find the value of
.
x that makes a II b.
3.
4.
a
b
5.
a --~:7--
b
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__
~
_
Geometry
Practice Workbook
With ExampleS
NAME~.
~~
__ ~
~
~
DATE
Practice with. Examples'
For use with pages 157-164
Use properties of parallel lines and construct parallel lines using
straightedge and compass
=l
VOCABULARY·
.
Theorem 3.11 If two lines are parallel to the same line, then they are'
i parallel to each other.
,
Theorem 3.12 In a plane, if two lines are perpendicular to the same
.Lline, then
. they are parallel to each other,
.
I
I
.......J
"
Showing Lines are Parallel
Explain how you would show that k
a.
k
II f.
b.
n
a. Because the 40° angle and angle I form a linear pair, mL 1 must equal 140°,
Thus L 1 and the other 140° angle are congruent. Because they are also
corresponding angles, lines k and
Converse postulate,
e are parallel
by the Corresponding Angles
b. Because the three angles with measures of 6xo, 5~0, and Xo form a straight
line, their sum must be 180°. So 6x + 5x + x = 180, Thus 12x = 180, and
therefore x = 15. We can now conclude that the angle with the 6xo measure is
a right angle [6 . (15) = 90]. Therefore, line n is perpendicular to line f.
Since line k is also perpendicular to line n (the 90° angle is indicated), lines k
and are parallel by Theorem 3,12.
e
!.
Geovneti'Y
Practice VVorkbook with Examples
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LESSON
NAME
_
DATE
Practice with Examples
CONTiNUED
-For use with pages 157-164
.~~~!.~~~I!.~.
!.~L~l!.I!.r!!.l!.~~.!.
Explain how you would show
1.
.~
2.
~
kill.
k
f
3. f
Naming Parallel Lines
Determine which lines, if any, must be parallel.
a
Lines c arid d are parallel because they have congruent corresponding angles.
Likewise, lines d and e are parallel because they have congruent corresponding
angles. Also, lines c and e are parallel because they are both parallel to the same
line, line d. Because mL 1 = 105° (L 1 and the 75° angle form a linear pair), L I
and the 100° angle are not congruent. Since L 1 and the 100° angle are
corresponding angles that are not congruent, lines a and b are not parallel.
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Geometoy
Practice Workbook with Examples
.
m:::z;s'?t'f'=on
LESSON
3.5
••
NAME
efEH
ft
_
DATE
Practice with Examples
For use with pages 157-164
.~
.~~~!.~~~l!.~.
!.l!.~.~1!.f!.I!!J!.~I!..?
.
Determine which lines, if any, must be parallel.
4.
b
5.
h
c
d
9
6.
Geometry
Practice Workbook
with Examples
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,
NAME
_
DATE
:--
_
Practice with Examples
For use with pages 165-171
Find slopes of lines and use slope to identify parallel lines in a coordinate
plane and write equations of parallel lines in a coordinate plane
Postulate 17 Slopes of Parallel Lines In a coordinate plane, two
nonvertical lines are parallel if and only if they have the same slope.
Any two vertical lines are parallel.
~
finding the Slope of a Line
Find the slope of the line that passes through the points (3, - 3) and (0, 9).
SOLUTION
Let (Xl' Yl)
= (3, -3) and (x2' h) = (0,9).
9 - (- 3)
0-3
12
-3
=
-4
The slope of the line is - 4.
Exercises for
...................
, Example
1
~
Find the slope of the line that passes through the given points.
,
1. (4,2) and (6,8)
2. (-3, -1) and (- 5, -11)
3. (;- 8, 12) and (0, -12)
4. (8,3) and (14,5)
5. (-7, -5) and (5, 4)
6. (-18,5)
and (4, 5)
i
·i
I'
I
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Geometry
Practice Workbook
with Examples
.
NAME
_
DATE
Practice with Examples
For ase with pages 165-171
Find slopes of lines and use slope to identify parallel lines in a coordinate
plane and write equations of parallel lines in a coordinate plane
VOCAiSUIi..ARY
Postulate 17 Slopes of Parallel Lines In a coordinate plane, two
nonverticallines are parallel if and only if they have the same slope.
Any two vertical lines are parallel.
Find the slope of the line that passes through the points (3, - 3) and (0, 9).
SOLUTION
Let (xl' Yl)
= (3, - 3) and (x2' h) = (0,9).
m= Y2 - YI
X2
-
Xl
9 - (- 3)
0-3
12
-3
=
-4
The slope of the line is - 4.
Exercises tOIT Example 1
. .
.............................................................•.........................................................................
Find the slope of the line that passes through the given points.
1. (4,2) and (6, 8)
2. (- 3, -1) and (- 5, -11)
3. (;- 8,12) and (0, -12)
4. (8,3) and (14,5)
5. (-7,
6. (-18,5)
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-5) and (5,4)
and (4, 5)
Geometry
Practice
Workbook with Examples
_
NAME
DATE
Practice with Examples
CONTINUED
For use with pages 165-171
Ide'ntiifvilffl
Parallel
lines
Find the slope of each line. Is a
a
'YV
/
/
/
1/1
(-5( 0) /! i /
J/r
/
0
i/
1/ I -1 )
/1
/
-<
lib?
/l(O,5)
b
;¥~
(0,2)1
-+--
!
.. _c __c·__
x
·····
Find the slope of a. Line a passes through (- 5; 0) and (0, 5).
5 - 0
lna
5
= 0 - (- 5) = "5 = 1
Find the slope of b. Line b passes through (- 2, 0) and (0, 2).
2 - 0
mb
2
= 0 - (- 2) = "2 = 1
Compare the slopes. Because a and b have the same slope, they are parallel.
Exercises for Example 2
•••••••••••••••••••••••.•••.•••••••••••••
~ ••••••••••••••••••••.•••••••.•••.•••.••••.•••.•••••.••••••••
~ •••.•••••••••••.•.
" •••.•.•••••.•••••.•••••.•••.•.•.••••
"
•••••.•.•••••••••••••••.•••••.••••
e •• c.'
••.•.•.•.•..•
e •..•.••..•.•.•••.•.••..••.•.•..•••
Find the slope of each line. Which lines are parallel?
8.
Y
,
I
h7~ 2)
i(5, 1)
1-
i !
x'
1
i
-( -7, ! -
(
1)
~)
i
!
I.~.~.
I
Geometry
Practice Workbook with Examples
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All rights reserved.
LESSON
CONTINUED
NAME_.~
_
DATE
Practice with Examples
For use with pages 165-171
,Writing an Equation of a Parallel Line
Line k has the equation y = - x - 4.
Line
.e is parallel
to kand passes through the point (1, 5). Write an equation of
.e.
SOLUTION
Find the slope. The slope of k is - 1. Because parallel lines have the same slope,
the slope of .e is also - 1.
Solve for b. Use (x, y)
= (1,5) and m. = -1.
y=mx+b
5=-1(1)+b
I
5=-I+b
i
Write an equation. Because m
I:
I
I
t
I
6=b
= -
1 and b
=
6, an equation
of
.e is y = - x + 6.
.~~.f!!.~
~~~~.
t~.~.
~l!.l!.'!!.f!.~i!.
.~
.
Write an equation of the line the passes through the. given point P
and is parallel to the line with the given equation.
9. P(10, 3), y
= x - 12
f
I
I
I
I
10.
p( - 5,2), y = -x - 9
11. PC-I, 2),y
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2
=
'3x -
2
Geometry.
Practice Workbook with Examples
NAME
DATE
~------------------------~--
Practice with Examples
For use with pages 172-178
Use slope to identify perpendicular lines in a coordinate plane and write
equations of perpendicular lines
VOCABULARY
Postulate 18 Slopes of Perpendicular Lines In a coordinate plane, two
nonverticallines are perpendicular if and only if the product of their
slopes is - 1.
Deciding Whether Lines are Perpendicular
~
~
a .. Decide whether PQ and QR are perpendicular.
b. Decide whether the lines are perpendicular .
•
Line f: 2x - 3y
=
-4
Line k: 3x
+
2y
=
3
SOLUTiON
a. Find each slope.
~
Slope of PQ
=
3 - 0
0 - (- 4)
3
=4
~
0- 3
-3
Slope of OR = -=
= -3
1- 0
1
Multiply slopes to see if the lines are perpendicular.
l.
4
(-3) = -~
4
.
~
~
The product of the slopes is not - 1. SO, PQ and QR are not perpendicular.
~-Geometry
~,c~Practice Vvorkbook with Examples
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l
--1&
LESSON
~
I
NAME
~~
_
DATE
Practice with Examples
CONTINUED
For use with pages 172-178
b. Rewrite each equation in slope-intercept form to find the slope.
e.
Line
Y
=
Line k:
2
4
3x + 3
slope
y
2
=-
=
slope
3
3
-"2x + "2
= --
3
3
2'
Multiply the slopes to see if the lines are perpendicular.
G) . ( -1)
= -
1, so the lines are perpendicular .
.~~.i!.~i?~~i!.~.
!.~!.
.~Jf.'!.'!!l!/i!..!
,
Decide whether lines k and
e
are perpendicular.
1. k passes through (3, 2) and (- 1, 5)
e passes
I
i
through (0,2) and (3, 6)
2. k has the equation 2x - 4y
e has the equation
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x
+
2y
= -3
= - 6
Geometry
Practice Workbook
vvith Exqmples
.
LESSON
NAME
_
DATE
_
Practice with Examples
CONTINUED
For use with pages 172-178
Wiitifl!j the Equation of a Perpendicular Line
Line k has equation y
=
~x - ~. Find an equation of line f that passes
through P(3, -1) and is perpendicular to k.
SOU.JTION
First determine the slope of
their slopes must equal - 1.
Ink'
2
3
-.
me
e. For k and e to be perpendicular,
the product of
= -1
me = -1
me = --
.
Then use In
3
2
3
=
-2 and (x, y)
=
(3, -1) to find b.
y=mx+b
3·
2
-1 = -- . (3) + b
2=b
2
.
S0, an equation
0
f
o :
-(.IS
Y
=
-lX3
7
+2
.~~~!.~~~I!.~.!.l!.~.~Jf.l!.'!!P..~I!..?
.
Line j is perpendicular to the line with the given equation and
line j passes through P. Write an equation of line j.
3. 4x
+ 7y
=
13,P(-2,6)
4.
5x - 2y = 3,P(O,
-~)5.
x
+
5y
=
6,P(-1,2)
~
..~.".'.-:~.
..•
~.'
Geometry
PIrlMir.R Workbook with Examples
Copyright © McDougal.Littell Inc.
All rights reserved.
LESSON
3.7
CONTINUED
NAME
_
DATE
_
Practice with Examples
For use with pages 172-178
_Writin.g
,
=
the Equation
of a,Perpendicular Line
Line k has equation y = ~x -
r
Find an equation of line f that passes'
through P(3, - 1) and is perpendicular to k.
SOLUTION
,
First determine the slope of .e. For k and
their slopes must equal - 1.
mk
•
2
3
-.
.e to be
perpendicular, the product of
rne = -1
m
e
= -1
,
3
me =--
2
,
Then use m
3
=
-'2 and
(x, y)
= (3, - 1)
to find b.
y =mx + b
_l . (3) + b
-1 =
2
2=b
2
.
S 0, an equation
f ()'
0 '(, IS Y =
-'23x + '27
.~~.~!.~~~'!.~.
!.C?~.~'!.l!.'!!.1?~'!..?
.
line j is perpendicular to the line with the given equation and
line j passes through P. Write an equation of line j.
3. 4x
+ 7y
= 13,P(-2,6)
Ge!timetry ,
Practice Workbook with Examples .
4. 5x - 2y = 3,P(O,
-%)
,5. x
+ 5y
= 6,P(-1,2)
Copyright ©McDougal Littell Inc.
All rights reserved,
I