P.o.D.
1.) Write an expression for the
nth term of
3
4
5
6
7
, , , , ,…
1! 2! 3! 4! 5!
2.) The fifth term of an
arithmetic sequence is 5.4 and
the 12th term is 11.0. Find the
nth tern. (in y=mx+b or a=nx+b)
3.) Expand (𝑥 + 2𝑦)4
4.) Find the coefficient of the
term 𝑎3 𝑏5 in the expansion of
(2𝑎 − 3𝑏)8 .
1.)
𝑛+2
𝑛!
2.) 𝑎𝑛 = 0.8𝑛 + 1.4
3.) 𝑥 4 + 8𝑥 3 𝑦 + 24𝑥 2 𝑦 2 + 32𝑥𝑦 3 +
16𝑦 4
4.) -108,864
10.1 – Lines
Learning Target: I can find the
inclination of a line; find the
angle between two lines; find
the distance between a point
and a line.
Inclination:
- The positive angle measured
counterclockwise from the xaxis.
(draw a picture on the
whiteboard)
Inclination and Slope:
- If a line has inclination, then
its slope is 𝑚 = tan 𝜃
EX: Find the inclination of the
line given by 5x-y+3=0.
We first need to find the slope of
this line. Remember, in standard
form, slope is
−𝐴
𝐵
.
−5
𝑚=
=5
−1
We can now use this slope to
find our angle of inclination.
𝑚 = tan 𝜃 →
5 = tan 𝜃 → tan−1 5 = 𝜃
→ 78.69° 𝑜𝑟 1.373 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 𝜃
Avoiding Common Errors:
Students sometimes forget that
the angle of inclination is always
considered positive. If a line has
a positive slope, then
0° < 𝜃 < 90°. If a line has negative
slope, then 90° < 𝜃 < 180°. In
order to find the angle with a
negative slope, remember that
when you use tan−1 on the
calculator you will need to add 𝜋
radians or 180 degrees to get the
result.
EX: Find the inclination of the
line 3x+7y=11.
−3
𝑚=
7
−3
tan 𝜃 = 𝑚 → tan 𝜃 =
→𝜃
7
−3
−1
= tan
= −23.1986°
7
𝜃 = −23.1986° + 180° = 156.801°
The Angle Between Two Lines:
- Two lines either intersect or
are parallel.
- Find the angle of inclination
for each line. Then find their
difference.
- tan 𝜃 = |
𝑚2 −𝑚1
1+𝑚1 𝑚2
|
EX: Find the angle between the
lines 2x+y=4 and x-y=2.
Begin by find the slope of each
line.
−2
𝑚1 =
= −2
1
−1
𝑚2 =
=1
−1
Now apply the formula.
1 − (−2)
3
tan 𝜃 = |
|=| |=3
1 + (−2)(1)
−1
𝜃 = tan−1 3 = 71.565°
*Let’s write a calculator
program to find the angle
between two lines.
EX: Use your program to find
the angle between 2x-y-4=0 and
3x+4y-12=0.
About 1.391 radians or 79.70
degrees.
The Distance Between a Point
and a Line:
If given a point (x,y) and the line
Ax+By+C=0, then the shortest
distance between them is given
by 𝑑 =
|𝐴𝑥+𝐵𝑦+𝐶 |
√𝐴2 +𝐵2
EX: Find the distance between
(0,2) and 4x+3y=0.
𝑑=
|4(0) + 3(2) + 0|
6
= = 1.2
5
√16 + 9
*Let’s write a program to find
the distance between a point
and a line.
EX: Use your program to find
the distance between (4,1) and
y=2x+1
2x-y+1=0 or -2x+y-1=0
8
≈ 3.58
√5
EX: Consider a triangle with
vertices A(0,0), B(1,5), and
C(3,1).
a.) Find the altitude from the
vertex B to AC.
We need to find the equation of
the line through AC.
1−0 1
𝑚=
=
3−0 3
1
1
𝑦 = 𝑥 + 𝑏 → 0 = (0) + 𝑏 → 0 = 𝑏
3
3
1
1
→𝑦 = 𝑥 →− 𝑥+𝑦 =0
3
3
→ −1𝑥 + 3𝑦 = 0
Now we can use the program for
the distance between a point
and a line.
𝑑=
|−1(1) + 3(5) + 0|
=
14
√1 + 9
√10
7√10
=
5
b.) Find the area of the
triangle.
Use the distance formula for
each side. Then apply Herron’s
formula for the area of a
triangle.
𝑑(𝐴𝐵) = √52 + 12 = √26
𝑑(𝐴𝐶 ) = √9 + 1 = √10
𝑑 (𝐵𝐶 ) = √4 + 16 = √20
After this lesson, you should be
able to:
1. Find an angle of inclination
2. Find the angle between any
two lines
3. Find the shortest distance
between a line and a point.
For more information, visit
http://www.mcg.net/nelson/CHAT/math/pr
ecalculus/overheads/sec.%2010.1.pdf
HW Pg.732 3-51 3rds, 62-76E