Transcript Slopes of Parallel and Perpendicular Lines LESSON 3-7 Additional Examples Line 1 contains P(0, 3) and Q(–2, 5).

Slopes of Parallel and Perpendicular Lines
LESSON 3-7
Additional Examples
Line 1 contains P(0, 3) and Q(–2, 5). Line 2 contains
R(0, –7) and S(3, –10). Are lines 1 and 2 parallel? Explain.
Find and compare the slopes of the lines.
Slope of line
Slope of line
y2 – y1
5–3
y2 – y1
–10 –(–7)
3–0
1
= x –x =
–2 – 0
2
1
2
= x –x =
2
1
=
2
–2
= –1
= –3 = –1
3
Each line has slope –1.
The y-intercepts are 3 and –7.
The lines have the same slope and different y-intercepts, so they are parallel.
Quick Check
HELP
GEOMETRY
Slopes of Parallel and Perpendicular Lines
LESSON 3-7
Additional Examples
Are the lines y = –5x + 4 and x = –5y + 4 parallel? Explain.
The equation y = –5x + 4 is in slope-intercept form.
Write the equation x = –5y + 4 in slope-intercept form.
x = –5y + 4
x – 4 = –5y
Subtract 4 from each side.
–1x+ 4 =y
5
Divide each side by –5.
5
y=–1x+ 4
5
5
The line x = –5y + 4 has slope – 1 .
5
The line y = –5x + 4 has slope –5.
The lines are not parallel because their slopes are not equal.
HELP
Quick Check
GEOMETRY
Slopes of Parallel and Perpendicular Lines
LESSON 3-7
Additional Examples
Write an equation in point-slope form for the line
parallel to 6x – 3y = 9 that contains (–5, –8).
Step 1: To find the slope of the line, rewrite the equation
in slope-intercept form.
6x – 3y = 9
–3y = –6x + 9
y = 2x – 3
Subtract 6x from each side.
Divide each side by –3.
The line 6x – 3y = 9 has slope 2.
Step 2: Use point-slope form to write an equation for the new line.
y – y1 = m(x – x1)
y – (–8) = 2(x – (–5))
y + 8 = 2(x + 5)
HELP
Substitute 2 for m and (–5, –8) for (x1, y1).
Simplify.
Quick Check
GEOMETRY
Slopes of Parallel and Perpendicular Lines
LESSON 3-7
Additional Examples
Line 1 contains M(0, 8) and N(4, –6). Line 2 contains
P(–2, 9) and Q(5, 7). Are lines 1 and 2 perpendicular? Explain.
Step 1: Find the slope of each line.
m1 = slope of line
m2 = slope of line
y –y
1
= x2 – x1 =
2
1
–6 – 8
4–0
y –y
7–9
2
= x2 – x1 =
5 – (–2)
2
1
=
–14
7
=–
4
2
=
–2
=–2
7
7
Step 2: Find the product of the slopes.
m1 • m2 = –
7
2
•– =1
2
7
Lines 1 and 2 are not perpendicular because the product
of their slopes is not –1.
HELP
Quick Check
GEOMETRY
Slopes of Parallel and Perpendicular Lines
LESSON 3-7
Additional Examples
Write an equation for a line perpendicular to
5x + 2y = 1 that contains (10, 0).
Step 1: To find the slope of the given line, rewrite the equation
in slope-intercept form. 5x + 2y = 1
2y = –5x + 1
Subtract 5x from each side.
y = – 5x + 1
Divide each side by 2.
2
2
The line 5x + 2y = 1 has slope – 5 .
2
Step 2: Find the slope of a line perpendicular to 5x + 2y = 1.
Let m be the slope of the perpendicular line.
– 5 m = –1
The product of the slopes of perpendicular lines is –1.
2
m = –1 • ( – 2 )
Multiply each side by – 2 .
m= 2
5
HELP
5
Simplify.
5
GEOMETRY
Slopes of Parallel and Perpendicular Lines
LESSON 3-7
Additional Examples
(continued)
Step 3: Use point-slope form, y – y1 = m(x – x1), to write an equation
for the new line.
y – 0 = 2 (x – 10)
5
y = 2 (x – 10)
5
Substitute 2 for m and (10, 0) for (x1, y1).
5
Simplify.
Quick Check
HELP
GEOMETRY
Slopes of Parallel and Perpendicular Lines
LESSON 3-7
Additional Examples
The equation for a line containing a lead strip is y = 1 x – 9.
2
Write an equation for a line perpendicular to it that contains (1, 7).
Step 1: Identify the slope of the given line.
1
y = 2x – 9
slope
Step 2: Find the slope of the line perpendicular to the given line.
Let m be the slope of the perpendicular line.
1
m = –1
2
m = –2
HELP
The product of the slopes of perpendicular lines is –1.
Multiply each side by 2.
GEOMETRY
Slopes of Parallel and Perpendicular Lines
LESSON 3-7
Additional Examples
(continued)
Step 3: Use point-slope form to write an equation for the new line.
y – y1 = m(x – x1)
y – 7 = –2(x – 1)
Substitute –2 for m and (1, 7) for (x1, y1).
Step 4: Write the equation in slope-intercept form.
y – 7 = –2(x – 1)
y – 7 = –2x + 2
y = –2x + 9
Use the Distributive Property.
Add 7 to each side.
Quick Check
HELP
GEOMETRY