M2 GEOMETRY – PACKET 4 FOR UNIT 2 – SECTIONS 3-3 AND 3-4
NAME
DATE
M2 Geometry – Assignment sheet for Unit 2 Lines and Angles, Packet 4
Unit 2 includes the following sections: 1-4, 1-5, 2-8, 1-6, 6-1, 3-1 to 3-6
Due
#
Assignment
Topics
2J
p. 193-194
# 12-15 all, 19-25 odd, 35, 38, 41, 43
3-3: Vocabulary: slope
Find slope of a line using formulas or graphs
p. 193-194
# 28-36 even, 47, 48
3-3:
Identify slopes of parallel or perpendicular lines
Use slope to determine whether two lines are
parallel, perpendicular, or neither
Graph lines that are parallel or perpendicular to
a given line
Pages 10-11 in this packet
3-4: Vocabulary: slope-intercept form, pointslope form, x-intercept, y-intercept
Find equations of lines in slope-intercept form
when the y-intercept is easily identifiable
Find equations of lines in point-slope form in all
other cases
Write equations of horizontal and vertical lines
Write equations of lines that are parallel or
perpendicular to a given line
2K
2L
Quiz on 3-3 and 3-4
1
M2 GEOMETRY – PACKET 4 FOR UNIT 2 – SECTIONS 3-3 AND 3-4
3-3
SLOPE OF A LINE
3-3
The slope of a line describes the direction and steepness of a line. The slope m of a line
rise y2  y1
containing points  x1 , y1  and  x2 , y2  is given by the formula m 
.

run x2  x1
1. Sketch an example of a line that has each kind of slope:
a. positive slope
b. negative slope
c. slope = 0
d. slope is undefined
2. Find the slope of each line.
a. AB
b. CD
c. EM
d. AE
e. EH
f. MB
2
M2 GEOMETRY – PACKET 4 FOR UNIT 2 – SECTIONS 3-3 AND 3-4
3. Sketch a graph of each line, and find its slope. Express answers as whole numbers or
fractions in simplest form.
a. L 1, 2  , M  6,3
b. P  1, 2  , Q  9,6 
c. T 1, 2  , U  6, 2 
d. V  2,10  , W  4, 3
3
M2 GEOMETRY – PACKET 4 FOR UNIT 2 – SECTIONS 3-3 AND 3-4
3-3
SLOPES OF PARALLEL AND PERPENDICULAR LINES
Check steps off as you complete them.
Point
Tool
3-3
Line
Tool
______Open Geogebra. Use the Line Tool to
make any line AB .
______Make point C not on AB .
______Select the Parallel Line tool. Follow the directions on the screen to make a line through
C parallel to AB .
______Use the Slope tool to find the slope of each line.
______Click on the Selection Arrow in the upper left
corner of the screen, and drag point A to change
the slope of both lines.
What do you notice about the slopes of both lines?
Complete this statement:
If two lines are parallel, then their slopes _________________________________________.
4
M2 GEOMETRY – PACKET 4 FOR UNIT 2 – SECTIONS 3-3 AND 3-4
Check steps off as you complete them.
______Click on the Undo button until you only have AB and
point C on the graph.
______Select the Perpendicular Line tool. Follow
the directions on the screen to make a line
through C perpendicular to AB .
______Use the Slope tool to find the slope of each line.
______Double-click over the name of each slope on the
left side of the screen, and change the names to
m1 and m2 as shown at right. (Your slopes don’t
have to match mine.)
______In the next available space below the slopes, type m1*m2, and press the Enter key on
your keyboard.
______Click on the Selection Arrow in the upper left corner of the screen, and drag point A to
change the slope of both lines.
What do you notice about the value of m1*m2?
Complete this statement:
If two lines are perpendicular, then their slopes _________________________________________.
5
M2 GEOMETRY – PACKET 4 FOR UNIT 2 – SECTIONS 3-3 AND 3-4
The converses of these rules are also true.
If two lines have the same slope, then they are parallel.
If two non-vertical lines are perpendicular, then the product of their slopes is –1.
1. Find the slopes of MN and RS , and determine whether they are parallel, perpendicular,
or neither.
a. M(0, 3), N(2, 4), R(2, 1), S(8, 4)
b. M(–1, 3), N(0, 5), R(2, 1), S(6, –1)
2. Graph the line that satisfies each condition.
a. Line passes through H(8, 5) and is perpendicular to AG
with A(–5, 6) and G(–1, –2).
b. Line m passes through C(–2, 5) and is parallel to LB with
L(2, 1) and B(7, 4).
6
M2 GEOMETRY – PACKET 4 FOR UNIT 2 – SECTIONS 3-3 AND 3-4
3-4
EQUATIONS OF LINES
3-4
Part 1: Review of Equations of Lines
You can write an equation of a line if you are given any of the following:
 the slope m and the y-intercept b,
 the slope m and a point  h, k  on the line, or

two points  x1 , y1  and  x2 , y2  on the line.
If m is the slope, b is the y-intercept, and  h, k  is a point on the line, then:

slope-intercept form is y  mx  b . Use when you already know the y-intercept.

point-slope form is y  k  m  x  h  . Use this when the y-intercept is not obvious.
1. Graph a line with the given characteristics. Determine whether slope-intercept or pointslope form should be used, and write an equation.
a. slope 2, y-intercept –3
b. slope –2, point (4, –2)
i.
i.
ii. slope-intercept or point-slope?
ii. slope-intercept or point-slope?
iii. equation:
iii. equation:
7
M2 GEOMETRY – PACKET 4 FOR UNIT 2 – SECTIONS 3-3 AND 3-4
1. (continued) Graph a line with the given characteristics. Determine whether slopeintercept or point-slope form should be used, and write an equation.
1
d. slope  , y-intercept 4
2
c. points (–2, –3) and (3, –5)
i.
i.
ii. slope-intercept or point-slope?
ii. slope-intercept or point-slope?
iii. equation:
iii. equation:
Part 2: Equations of Horizontal and Vertical Lines
Horizontal and vertical lines have special equations. You should not have to use slopeintercept form or point-slope form to write their equations.
2. Graph the horizontal line passing through (3, 2).
a. Find the coordinates of 3 other points on this line.
b. What do these points have in common? Write an
equation describing this.
8
M2 GEOMETRY – PACKET 4 FOR UNIT 2 – SECTIONS 3-3 AND 3-4
3. Graph the vertical line passing through (3, 2).
a. Find the coordinates of 3 other points on this line.
b. What do these points have in common? Write an
equation describing this.
4. Complete each statement:
a. The horizontal line passing through the point (h, k) has equation ________________.
b. The vertical line passing through the point (h, k) has equation___________________.
5. Write equations for these lines:
a.
b.
c. the line through (5,2) and (5,9)
d. the line through (3,1) and (8,1)
9
M2 GEOMETRY – PACKET 4 FOR UNIT 2 – SECTIONS 3-3 AND 3-4
ASSIGNMENT 2L
1. Graph a line with the given characteristics. Determine whether slope-intercept or point-slope form should be
used, and write an equation.
3
b. slope  , y-intercept 1
4
a. points (–1, –3) and (3, –5)
i.
i.
ii. slope-intercept or point-slope?
ii. slope-intercept or point-slope?
iii. equation:
iii. equation:
d. x-intercept 4, y-intercept 2
c. slope 2, point (–5, –3)
i.
i.
ii. slope-intercept or point-slope?
ii. slope-intercept or point-slope?
iii. equation:
iii. equation:
Assignment 2L continues on the next page 
10
M2 GEOMETRY – PACKET 4 FOR UNIT 2 – SECTIONS 3-3 AND 3-4
2. Graph a line with the given characteristics. Determine whether slope-intercept or point-slope form should be
used, and write an equation.
a. passes through (–1, 5) and
1
is perpendicular to y  x  3
2
b. passes through (1, 3) and
2
is parallel to y   x  1
3
i.
i.
ii. slope-intercept or point-slope?
ii. slope-intercept or point-slope?
iii. equation:
iii. equation:
3. Determine whether the lines are parallel, perpendicular, or neither.
1
3
a. y   x  12, y  3x  7
b. y  4  2  x  5 , y  3  2  x  1
c. y  3  6  x  2  , y  x  6
d. y   x  5, y  1   x  4 
2
3
11
3
2
M2 GEOMETRY – PACKET 4 FOR UNIT 2 – SECTIONS 3-3 AND 3-4
Practice for Sections 3-3 and 3-4
Find the slope of each line. Express answers as whole numbers or fractions in simplest form.
1. G(–2, 5), H(1, –7)
2. J(–5, –2), K(5, –4)
3.
4.
Find the slopes of AB and MN , and determine whether they are parallel, perpendicular, or neither.
5. A(–1, 4), B(2, –5), M(–3, 2), N(3, 0)
6. A(–4, –8), B(4, –6), M(–3, 5), N(–1, –3)
Graph the line that satisfies each condition.
7. passes through Y(3, 0), parallel to ⃡𝐷𝐽
with D(–3, 1) and J(3, 3)
8. passes through T(0, –2), perpendicular
to ⃡𝐶𝑋 with C(0, 3) and X(2, –1)
12
M2 GEOMETRY – PACKET 4 FOR UNIT 2 – SECTIONS 3-3 AND 3-4
Graph a line with the given characteristics. Determine whether slope-intercept or point-slope form
should be used, and write an equation.
9. slope –4, y-intercept 3
10. Points (5,2) and (1,6)
i.
i.
ii. slope-intercept or point-slope?
ii. slope-intercept or point-slope?
iii. equation:
iii. equation:
11. slope –3, point (2, –4)
12. slope
i.
2
, point (0, –6)
5
i.
ii. slope-intercept or point-slope?
ii. slope-intercept or point-slope?
iii. equation:
iii. equation:
13
M2 GEOMETRY – PACKET 4 FOR UNIT 2 – SECTIONS 3-3 AND 3-4
Write an equation for each line shown or described.
13. the line parallel to line r that contains (1, –1)
14. the line perpendicular to line s that contains (0, 0)
15. x-intercept is –2, y-intercept is –1
16. passing through (6,8) and (6,10)
17. passing through (5,7) and (21,7)
18. the vertical line passing through (11,9)
19. the line with x-intercept 12 and slope
3
4
20. the horizontal line passing through (11,9)
14
M2 GEOMETRY – PACKET 4 FOR UNIT 2 – SECTIONS 3-3 AND 3-4
Review of 3-3 and 3-4
Determine the slope of the line that contains the given points.
2. I(–2, –9), P(2, 4)
1. B(–4, 4), R(0, 2)
Find the slope of each line in the figure at right.
3. LM
4. GR
5. a line parallel to GR
6. a line perpendicular to PS
Determine whether KM and ST are parallel, perpendicular, or neither. Graph on graph paper to verify
your answer.
7. K(–1, –8), M(1, 6), S(–2, –6), T(2, 10)
8. K(–5, –2), M(5, 4), S(–3, 6), T(3, –4)
Graph the line that satisfies each condition.
1
9. slope =  , contains U(2, –2)
2
10. slope =
15
4
, contains P(–3, –3)
3
M2 GEOMETRY – PACKET 4 FOR UNIT 2 – SECTIONS 3-3 AND 3-4
Graph the line that satisfies each condition.
⃡
11. contains B(–4, 2), parallel to 𝐹𝐺
with F(0, –3) and G(4, –2)
⃡
12. contains Z(–3, 0), perpendicular to 𝐸𝐾
with E(–2, 4) and K(2, –2)
Graph a line with the given characteristics. Determine whether slope-intercept or point-slope form
should be used, and write an equation.
13. slope
2
, point (0, –1)
3
14. slope
i.
3
, point (4, 6)
2
i.
ii. slope-intercept or point-slope?
ii. slope-intercept or point-slope?
iii. equation:
iii. equation:
16
M2 GEOMETRY – PACKET 4 FOR UNIT 2 – SECTIONS 3-3 AND 3-4
Graph a line with the given characteristics. Determine whether slope-intercept or point-slope form
should be used, and write an equation.
15. slope –4, y-intercept 3
16. Points (5,2) and (1,6)
i.
i.
ii. slope-intercept or point-slope?
ii. slope-intercept or point-slope?
iii. equation:
iii. equation:
17. parallel to the line shown below
and passing through (3,2)
18. perpendicular to the line shown below
and passing through (–2, –1)
i.
i.
ii. slope-intercept or point-slope?
ii. slope-intercept or point-slope?
iii. equation:
iii. equation:
17